Thanks to the LHC one can make particles collide and annihilate into other particles.
Therefore, currently, what is the mass of the most massive particle one could create theoretically given the capacities of the LHC.
Can we create dark-matter particles via the LHC?
The absolute maximum is the centre of mass energy, 14 TeV. But that's the energy of the colliding protons. In fact, outgoing particles are created through the collision of a constituent of one proton, and the constituent of the other proton. Constituent here means a quark or a gluon. The problem then becomes rather complex, involving what particle physicists call Parton Density Function, i.e. what is the probability to find a given type of parton carrying a given fraction of the proton momentum. But I think you will allow me to give a layman answer and say that introducing a factor 3 (for the 3 quarks the proton is mostly made of) is good enough! So that's a reduction of a factor 10 basically (I mean 9 = 10 at that level of handwaving!).
There is a lot of work on searching for Dark Matter particles at LHC, definitively! A very recent review can be found at https://arxiv.org/abs/1702.02430. This is a "pro" article, but I think you could get some useful info from the conclusion.
A more thorough answer requires to pay attention to those Parton Density Functions (PDF) I mentioned. Let's take an example: it is possible to find a quark up carrying 99% of the proton momentum but this is much less probable than finding one carrying close to 1/3 let's say. So in the absolute, 14 TeV is the maximum but collisions even approaching this would be extremely unlikely. Eventually, it becomes a probabilistic argument because for each maximum mass between 0 and 14 TeV, you have a probability assigned to it. When I wrote 1.4 TeV above, I therefore meant an approximation of the most likely maximum mass, across a vast number of collisions.
From there, if you wanted to see the distribution of probability, it becomes really involved as you have to consider all possible quark-quark, quark-gluon and gluon-gluon collisions. I ought to have the answer as I did my PhD on a very related subject but this was 20 years ago!
Short answer, the Top quark.
In Hadron collisions top quarks are produced in pairs through the processes $q \overline{q} \rightarrow t\overline{t}$ and $g \overline{g} \rightarrow t \overline{t}$ at leading order in QCD.
Cortiana's paper Top-quark mass measurements: review and perspectives should give you a little more information on the detection and measurements.
