I’m not saying it’s impossible to find functions with different limits. Just that they’re less common. If you have g(x)^{f(x)}, you need to make f(x) approach 0 logarithmically more slowly than g(x) to dilute its influence.
Saying it’s undefined is perfectly legitimate. But if we want to define it, 1 is probably the only sensible answer.
If you’re straight up asking what 0 ^ 0 equals as an expression devoid of context, it’s best left undefined because of the tension between considering it 0 (from 0^a) or 1 (from a^0).
If you have 0^0 in a limit, you might rearrange it into e ^ (0 * log (0)), in which case you then end up with a 0 * infinity indeterminate form in the exponent, but in that case it might be easier to see how to evaluate it.
If you have 0 ^ 0 as part of the output of a combinatorial formula, it usually means that you’re effectively counting the number of ways to put 0 items into 0 boxes, as stated earlier, and there’s exactly one way of doing that, because doing nothing is a thing that is always counted. There might be other areas of study in which you have formulas that might give you 0 ^ 0 for certain data sets, and in those fields it might be a bit more abstract what exactly it’s determining, but it’s probably safer to have it equal to 1 following from combinatorics, because that’s probably how the exponential that could be 0 got into your formula.
Unambiguously though, 0! = 1.
Another version of the same thing is that for (finite) sets A and B with a and b elements, resp., the number of functions A\to B is a^b. The number of functions of the empty set to itself is 1. The definition, if any, of 0^0 is going to depend on the context. In the context of set theory, 0^0=1. In the context of real variables, it is undefined.
No matter what you do, the function has a discontinuity at (0,0). A hole in a domain is a kind of discontinuity, after all. So it’s not enough to say “0^0 is undefined so we don’t have a discontinuity there”. We’re not deciding whether to have a discontinuity; we’re just deciding what kind of discontinuity to have.