Answer by freakish for Does the exponential of a function converge? What can we do with it?

I think $\exp(t)$ converges for every possible input of $t$, but I haven’t found a proof yet.

For all real and complex inputs of $t$, yes. This is not something trivial to show. You need proper background in set theory, how naturals, integers, reals and complex numbers work. You need to understand not only what it means for a series to converge, but what a series even is. Convergence tests are important as well. And how all of that works in complex numbers is even more complicated. You can start by reading some book on mathematical analysis, e.g. Robert Magnus "Fundamental Mathematical Analysis".

Depending on how much you already know this question might have an easy answer. For example we can apply d'Alembert's criterion to $a_n=\frac{t^n}{n!}$ to get

$$\lim_{n\to\infty}\bigg|\frac{a_{n+1}}{a_n}\bigg|=\lim_{n\to\infty}\frac{t}{n+1}=0$$and so the series $\sum_{n=0}^\infty a_n$ converges absolutely for each complex $t$. But still, my advice is to read some book.

Question: Does the exponential “converge” for function inputs?

Yes. And this is actually quite simple. It doesn't matter that the input is of the form $f(x)$, and that the particular series is the exponential function. As long as a series $S(t)=\sum_{n=0}^\infty \alpha(t, n)$ is well defined/convergent at some $y$, and $f(x)=y$ then $S(f(x))$ is well defined, in fact equal to $S(y)$.

In particular since $\exp(y)$ is defined for all real $y$, then $\exp(t^2)$ is also well defined, because $t^2$ is a real. So is $\exp(sin(t))$. And so is any $\exp(f(t))$ as long as $f(t)$ is a real/complex.

We also define $\exp(t)$ for complex numbers, matrices, and basically anything you want to plug in.

Not anything. That's a serious exaggeration. In order for this definition to make sense on a set $X$ you need the following structures on it:

1. You need to know how to add elements of $X$. Typically $X$ would be an abelian group.
2. You need to know how to multiply elements of $X$ by rationals, or at least rationals of the form $\frac{1}{n!}$. So scalar multiplication. Typically this means a vector space over rationals, or even reals.
3. You need to know how to multiply elements of $X$ with each other, or at least form powers $x^n$. Typically this means an associative algebra.
4. You need some form of convergence, e.g. a topology.

So this is an absolute minimum. But in order to prove things you will need more, e.g. those structures have to be related with each other. Additionally having a complete metric would be useful. In fact a proper setup for considering exponential functions is Banach algebra. Which covers all those examples you've mentioned. But it is a quite restrictive structure, and not used commonly.

So in my opinion its the opposite: there aren't many mathematical structures which you can use as inputs for the exponential function. For most of the time it is enough to consider reals and complex numbers only.