00 = 1 is what my HP calculator gets, and it seems reasonable.

x0 = 1 for all nonzero values of x.

(.0000001)0 = 1

(-.0000001)0 = 1

x0 = 1 as x approaches zero from either direction.

It would be strange for it to have some other value at zero.

In the complex plane, z0 = 1 for all nonzero values of z. This makes even even weirder for it to be nonzero at zero. Zero is surrounded in all directions by values for which z0 = 1

0 / 0 is undefined. The value depends on how you got the expression. For example consider the sin function.

sin(x) = x - x3 / 3! + x5 / 5! - x7 / 7! + . . .

From the above, sin(0) = 0 and

sin(x) / x = 1 - x2 / 3! + x4 / 5! - x6 / 7! + . . .

The above indicates that sin(x) / x = 1 for x = 0 and

2 * sin(x) / x = 2 for x = 0

The last two expressions are 0 / 0 if you merely substitute 0 for x.

I think you made a keying error in your exponential, or your calculator got confused due to ambiguity. My HP is a stack calculator, and does not evaluate algebraic expressions. I cannot key in your exponential and test it.

2^3^2 might be ambiguous. I am not sure which of the following it might be.

( 23 )2 = 64

( 2 )3^2 = 29, which is 512

I see no way to interpret it so that it evaluates to a power of 3.