For instance, I have signals such as
$$x(t)=e^{\cos(t)}$$
$$x(t)=t\cdot e^{\cos(t)}$$
How should I approach these kinds of signals to determine their fundamental periods?
The fundamental period is the smallest $p > 0$ for which $x(t) = x(t+p)$ holds.
For the fist $x(t)$, know that the exponential is a bijective function on the real line, so the period of $x(t)$ is the same as the period of $\cos(t)$, namely $2\pi$. For the second $x(t)$, there is no fundamental period because the function is not periodic. Any continuous periodic function must be bounded, but clearly the second choice of $x(t)$ is not.