If the Michelson-Morley experiment hadn't been conducted, are there any other reasons to think, from the experimental evidence available at that time, that Einstein could think of the Special Theory of Relativity?
Is there any other way to think why the speed of light is the ultimate speed limit?
A lot of people find it somewhat surprising, but Einstein's initial formulation of special relativity was in a paper, On the electrodynamics of moving bodies, that makes very little reference to the Michelson-Morley result; instead, it is largely based on the symmetry of electromagnetic analyses in different frames of reference.
From a more modern perspective, there is a strong theoretical case to be made that special relativity is, at the very least, a strong contender for the description of reality. These are beautifully summed up in Nothing but Relativity (doi), but the argument is that under some rather weak assumptions, which are essentially
you are essentially reduced to either
with no other options.
To get to reality, you need to supplement this theoretical framework with experiment - there's no other way around it. The Michelson-Morley experiment is, of course, the simplest piece of evidence to put in that slot, but in the intervening century we have made plenty of other experiments that fit the bill. From a purely mechanical perspective, the LHC routinely produces $7\:\mathrm{TeV}$ protons, which would speed at about $120c$ in Newtonian mechanics: it is very clear that $c$ is a universal speed limit, because we try to accelerate things faster and faster, but (regardless of how much kinetic energy they hold) they never go past $c$.
If you want something from further back, this is precisely the reason we developed the isochronous cyclotron in the late 1930s and then switched to synchrotrons back in the 1950s - cyclotrons require particles to keep in sync with the driving voltage, but if they approach the speed of light they can no longer go fast enough to keep up. We have upwards of eighty years of history of being able to mechanically push things to relativistic regimes.
If you wish for an answer inscribed within "experimental physics as of 1888, minus the Michelson-Morley result" then, as I said, the symmetry properties of electromagnetism (which are directly compatible with SR as derived from $v\ll c$ experiments, but require aether theories to make sense in galilean relativity) were plenty to convince Einstein that SR was the right choice.
Edit:
As pointed out in a comment, Einstein's original paper does make some reference to Michelson-Morley(-type) experiments, in his second paragraph:
Examples [like the reciprocal electrodynamic action of a magnet and a conductor], together with the unsuccessful attempts to discover any motion of the earth relatively to the “light medium,” suggest that the phenomena of electrodynamics as well as of mechanics possess no properties corresponding to the idea of absolute rest.
However, apart from this small nod, he makes no substantive references to the aether or its equivalents: the paper starts with the relativity postulates (based on the constancy of the speed of light), uses those to construct special relativity (as pertains transformations between moving frames, and so on), and then builds his case for it on the transformation properties of the equations of electromagnetism: these provide the deeper fundamental insight that underlies the symmetry of analysis of electromagnetic situations performed on different moving frames of reference.
In the modern day we have very direct measurements of the velocity behavior of particles as you add kinetic energy. The CEBAF accelerator at Jefferson Lab only works because adding (a lot!) more energy to the electrons after they have entered the racetrack for the first time doesn't change their speed relative the lab enough to be measured.
Really.
Between leaving the booster with 123 MeV kinetic energy and being delivered to the halls with up to 12 GeV kinetic energy (a factor of ~100 increase) the beam speed stays constant to high precision.
In calculations this means we go from $\gamma_\text{booster} = 241$ to $\gamma_\text{max} = 23500$. You'll need a high precision calculator to determine the difference in speeds implied by this and the real world bears such considerations out.
You just can't push a particle faster than $c$.
The Fizeau experiment is a way to measure the speed of light in a moving medium. From our modern standpoint, it provides an experimental test for the Lorentz transformation of a velocity $u$ in a reference frame moving at $v$, in the regime where $u$ is of order $c$ and $v \ll c$. Crucially, the apparatus is sensitive enough to discriminate the Lorentz transformation formula $u' = \frac{u + v}{1 + \frac{uv}{c^2}}$ from the Galilean formula $u' = u + v$. The experiment was conducted in 1851, which was 36 years before the Michelson-Morley experiment.
Fizeau's results were unexpected at the time, disagreeing with naive ether-drag theories. However, the response among theorists was to favour more convoluted ether-drag theories, where different materials dragged the ether to different extents. These theories grew even more complex to incorporate dispersion, i.e. different wavelengths of light having different refractive indices.
Here are the conclusions drawn by Fizeau (emphasis mine):
Either, first, the æther adheres or is fixed to the molecules of the body, and consequently shares all the motions of the body; or secondly, the æther is free and independent, and consequently is not carried with the body in its movements; or, thirdly, only a portion of the æther is free, the rest being fixed to the molecules of the body and, alone, sharing its movements.
...
I conclude, then, that [the first] hypothesis does not agree with experiment. We shall next see that, on the contrary, the third, or Fresnel's hypothesis, leads to a value of the displacement which differs very little from the result of observation.
...
The success of this experiment must, I think, lead to the adoption of the hypothesis of Fresnel, or at least to that of the law discovered by him, which expresses the relation between the change of velocity and the motion of the body; for although the fact of this law being found to be true constitutes a strong argument in favour of the hypothesis of which it is a mere consequence, yet to many the conception of Fresnel will doubtless still appear both extraordinary and, in some respects, improbable; and before it can be accepted as the expression of the real state of things, additional proofs will be demanded from the physicist, as well as a thorough examination of the subject from the mathematician.
On the Effect of the Motion of a Body upon the Velocity with which it is traversed by Light (1860) Hippolyte Fizeau, Philosophical Magazine, Series 4, vol. 19, pp. 245-260
This is becoming a little like a list question, but here's another way you can do it without light.
As Emilio Pisanty eloquently describes, there are very strong theoretical grounds, using only symmetry results that intuitive clear to all of us from a very young age (<10years), that there is some, unique, universal signalling speed limit that is also inertial-frame-invariant. We only need to measure this parameter $c$ to find the complete laws. Notice I said "parameter" rather than speed, because we don't needfully have to observe something moving at an inertial-frame-invariant speed to derive $c$ experimentally. This is because the these same arguments give us the full form of the Lorentz transformation (without the actual value of $c$). In particular, they give us the time-dilation factor $\gamma(v)$ as a function of the relative velocity $v$.
So we can use any experiment that observes $\gamma(v,\,c)$ as a function of $v$ and curve fit the experimental results to $\gamma(v,\,c) = 1/\sqrt{1-(v^2/c^2)}$ by adjusting the $c$ parameter for tightest fit of results to the theoretical curve. If our experiments include values of $v$ that are a significant fraction of $c$, then our estimate of $c$ will be a good one.
There is a whole list of experiments that could work in this way.
We could, for example, measure muon decay lifetime measurements $\tau = \tau_0 \,\gamma(v,\,c)$ as a function of $v$.
Or, we could do a lower energy variation on Dmckee's example and measure $v$ as a function of kinetic energy $E = m_o\,c^2 (\gamma(v,\,c) -1 )$ and curve fit this one. We would only have to measure up to about $v=c/2$ (when $\gamma = 1.155$) to get a pretty good estimate of $c$ with modestly accurate equipment.
Of course, the experimental results get better as our technology improves and our experiments access higher and higher speeds.
Eventually they reach the situation described by Dmckee: we simply can't push the particles faster and we're effectively at $c-\epsilon$ speeds, so we can read off $c$ really accurately.
But, supposing we had progressively worked towards better and better experimental results using higher and higher velocities as I described and no-one had ever decided the speed of light were frame-invariant. I should think the estimates one would get at $v=c/4$ (when $\gamma = 1.033$) would, with a large number of repeats and good statistical processing would give an estimate of $c$ accurate enough that someone would say, "Hey, I think I know something that actually moves at the speed parameter $c$"!
The strongest current experimental evidence is the standard model of particle physics, the beautiful symmetries of SU(3)xSU(2)xU(1) with the plethora of data that produced them, would fall on their face if c were not the limiting velocity, i.e. if special relativity did not hold.
Every single mass measurement in the particle data book , comes from using energy and momentum conservation equations based on the algebra of the four vectors of special relativity and thousands upon thousands of measured events .
Obviously:
The speed of whatever may serve for exchanging signals between participants is necessarily less, or at most as great as, the speed at which the signal front
(signal front speed) propagates between them;
provided, of course, a measure of "speed" is defined and applicable at all
(which in turn presumes or is at least associated with a suitable measure of "distance", and with a suitable characterization of the relevant participants as a system with geometric relations between each other, a "frame").
The identification of signal front speed $c_0$ with "speed at which electromagnetic wave(form)s propagate in vacuum" is foremost a definition of "vacuum" (in terms of refractive index $n$) and therefore related to how to define refractive index and to measure its values accordingly (where the question arises: did Michelson/Moreley even do that?);
and therefore of practical and historical relevance.
Edit (related to comments): It should be noted that the reasons described above are not derived from any experimental evidence (and therefore contrary to such a presumption expressed in the OP question), but they can and must be considered already in the definition of physical quantities (such as "speed", especially) and therefore already in the design of any possible experiments for measuring "speed".
Without the Michelson-Morley experiment, is there any other reason to think speed of light is the universal speed limit?
The M-M experiment proofs only that the earth's movement is not observable under the rotation of the used interferometer. What is proofed is the time of flight on different paths between light emission point and the observations screen. So the M-M experiment does not proof that the speed of light is the universal speed limit.
The question about the speed limit is answerable in an easy way. Suppose you are the owner of rockets which fly say 6.000 km/h. How do you think to which speed you can accelerate an ant as well as an elephant with such rockets? As long as the empirical facts say that EM radiation has the highest observed speed it could not be found any method to accelerate something to a speed greater the speed of light.
There are some other interesting points:
Anyway the question about other experiments for the proof of the absence of any aether is a very interesting. The M-M experiment has a weakness. Since the gravitational potential has an influence on the speed of light and since gravitational influences the space around the mass it has to be proofed that near the earths surface gravitational does not move the space in the same way as a rotating sphere in a liquid should not influence the liquid. In reality it does.
It is my understanding that the limit is simply asymptotic... as speed increases, so does mass, so increasing speed requires ever increasing energy, to account for the new inertia... eventually the amount of energy needed to accelerate would be infinite, that would the speed limit for all matter/energy anywhere in the universe
In 1887 (prior to FitzGerald and Lorentz advancing the ad hoc length contraction hypothesis) the Michelson-Morley experiment UNEQUIVOCALLY confirmed the variable speed of light predicted by Newton's emission theory of light and refuted the constant (independent of the speed of the light source) speed of light predicted by the ether theory and in 1905 adopted by Einstein as his second (constant-speed-of-light) postulate:
https://en.wikipedia.org/wiki/Emission_theory "Emission theory, also called emitter theory or ballistic theory of light, was a competing theory for the special theory of relativity, explaining the results of the Michelson–Morley experiment of 1887. [...] The name most often associated with emission theory is Isaac Newton. In his corpuscular theory Newton visualized light "corpuscles" being thrown off from hot bodies at a nominal speed of c with respect to the emitting object, and obeying the usual laws of Newtonian mechanics, and we then expect light to be moving towards us with a speed that is offset by the speed of the distant emitter (c ± v)."
http://philsci-archive.pitt.edu/1743/2/Norton.pdf "The Michelson-Morley experiment is fully compatible with an emission theory of light that CONTRADICTS THE LIGHT POSTULATE."
Banesh Hoffmann, Relativity and Its Roots, p.92: "There are various remarks to be made about this second principle. For instance, if it is so obvious, how could it turn out to be part of a revolution - especially when the first principle is also a natural one? Moreover, if light consists of particles, as Einstein had suggested in his paper submitted just thirteen weeks before this one, the second principle seems absurd: A stone thrown from a speeding train can do far more damage than one thrown from a train at rest; the speed of the particle is not independent of the motion of the object emitting it. And if we take light to consist of particles and assume that these particles obey Newton's laws, they will conform to Newtonian relativity and thus automatically account for the null result of the Michelson-Morley experiment without recourse to contracting lengths, local time, or Lorentz transformations. Yet, as we have seen, Einstein resisted the temptation to account for the null result in terms of particles of light and simple, familiar Newtonian ideas, and introduced as his second postulate something that was more or less obvious when thought of in terms of waves in an ether. If it was so obvious, though, why did he need to state it as a principle? Because, having taken from the idea of light waves in the ether the one aspect that he needed, he declared early in his paper, to quote his own words, that "the introduction of a 'luminiferous ether' will prove to be superfluous."
Relativistic effects have been tested (e.g time dialation) and the experiments agree with the theory. Ultimately this means that objects approaching c increase in mass to the point where you will need ever increasing amounts of additional energy to accelerate them further.
