(Note: I deliberately didn't write the title in Latex because the notation doesn't display properly with power towers. Specifically regarding the ellipsis, which displays as "..." instead of ".^{.}^{.}", and has caused confusion in past questions)
This might become a little confusing without the use of latex so what I'm going to do is define $n$ (where $n \in \mathbb{N}$) to equal the number of exponents in the sequence, so that when $n=1$, the sequence is $0^{(-1)}$. When $n=2$, the sequence is $0^{{(-1)}^{(-2)}}$ and so on.
At first I thought this would be a rather easy to answer and trivial question; any continuation of the sequence past $n=1$ should result in an answer equal to zero, which I tried to verify using Wolfram Alpha but it turns out to not be the case. In actuality the value of the sequence for varying values of $n$ is as follows;
$$n=1, \text{value } = \infty$$ $$n=2, \text{value } = 0$$ $$n=3, \text{value } = 0$$ $$n=4, \text{value } = \infty$$ $$n=5, \text{value } = 0$$ $$n=6, \text{value } = 0$$ $$n=7, \text{value } = \infty$$
...and so on. The alternating pattern in the values is particularly interesting, and has me wondering what the value will end up being as $n \to \infty$