From Misner, Thorne and Wheeler, page 115.
0-Form or Scalar, $f$
An example in the context of 3-space and Newtonian physics is temperature $T\left(x,y,z\right),$ and in the context of spacetime, a scalar potential, $\phi\left(t,x,y,z\right).$
I'm trying to think of an example of such a scalar potential. Is there one? Electrostatic potential is the time component of the electromagnetic 4-vector potential, so it's really a vector with 0-valued space components.
Within the Standard Model, the simplest model of the Higgs field is a multiplet of Lorentz scalar fields. This multiplet does have a non-trivial transformation under an underlying gauge group of the Standard Model; but under Lorentz transformations, the Higgs field is invariant, as all good Lorentz scalars should be.
Of course, the Higgs field is not a "potential" in the sense that a "potential" is a field whose derivative is a physically observable field; so if you're strictly looking for a scalar potential this isn't what you're looking for. But to the best of my knowledge it is the only fundamental scalar field we know to date.
If you have a vector or tensor field, then you can get a scalar field by contraction.
Examples:
$J^\mu$ = 4-flux of some quantity. Scalar field: $\rho = \sqrt{J^\mu J_\mu}/c$. Interpretation: proper density.
$k^\mu$ = 4-wave vector; $x^\mu$ = 4-position. Scalar field: $\phi = k^\mu x_\mu$. Interpretation: phase of a plane wave.
Electromagnetic field tensor $F^{\mu\nu}$. Scalar fields: $F^\mu_\mu$ and $F^{\mu\nu} F_{\mu \nu}$ and $F^{\mu \nu} \tilde{F}_{\mu \nu}$. The first of these is zero, the second is $2(E^2 - c^2 B^2)/c^2$ and the third is $4 {\bf E} \cdot {\bf B}$.
The above are scalar fields, though not normally called 'potentials' because their gradient does not relate to a force. However we can introduce a potential which is by definition a scalar invariant, and then consider the gradient to be a 4-force. We thus obtain $$ f^\mu = - \partial^\mu \phi. $$ Such a 4-force is not the electromagnetic force, but it can be used to construct simple models of the strong force.
There are two different things one can mean by a potential.
The first is in the sense of a gauge field whose derivatives (in some combination) give the field-strength tensor. For example, the electromagnetic potential $A^\mu(t,\vec{x})$ as you mention. One can certainly write down a theory with a scalar gauge field $B(t,\vec{x})$, however, such a gauge field does not appear in the standard model of particle physics. But, there is nothing wrong with writing down such a theory. See, for example, this paper where the $U(1)$ symmetry is gauged using a scalar gauge field.
The second is in the sense of the potential terms, i.e., the interaction/self-interaction terms in the Lagrangian density. For example, we say things like "the shape of the Higgs potential looks like the logo of a famous StackExchange site". All such potential terms are always scalar because a Lagrangian is not allowed to be charged.