Is $\exp(a\exp(a\exp(a\cdots)))$, where $a=\pi/2$, a valid representation of $i$? Take $i=e^{\frac{i\pi}{2}}$. For this question it will be more convenient to write it as $i=e^{\frac{\pi}{2}i}$. Substituting in this value for $i$, we get
$$i=e^{\frac{\pi}{2}e^{\frac{\pi}{2}i}}$$
For convenience, let $a=\frac{\pi}{2}$. Repeating this gives us
$$i=e^{ae^{ae^{ae^{\ldots}}}}$$
So, if we apply this substitution infinitely many times, does the $i$ in the end just ... go away? This is a bit confusing because you need an $i$ in there to collapse the whole expression, but things at infinity are weird.
If someone could help me better understand this representation that would be great.
 A: We can call $$\large{e^{\frac{\pi}{2}e^{\frac{\pi}{2}i}}}$$ a "power tower" of height $2$.
The "power tower"
$$\Large{e^{\frac{\pi}{2}e^{\frac{\pi}{2}e^{\frac{\pi}{2}\cdot^{\cdot^{\cdot^{e^{\frac{\pi}{2}i}}}}}}}}$$
of height $n$ ($n$ iterations) converges to $i$ as $\ n\to\infty\ $ because for each $\ n,\ $ the value of the tower always equals $\ i.$
On the other hand,
$$\Large{e^{\frac{\pi}{2}e^{\frac{\pi}{2}e^{\frac{\pi}{2}\cdot^{\cdot^{\cdot}}}}}}$$
which is different to the power tower limit just discussed, is not equal to a real number.
In general, doing an operation ($\ +, \times,\ $ ^ , other) an infinite amount of times is impossible to do all at once: you must perform the operation a finite amount of times and then let the "number of times" that you perform the operation increase without bound. If the sequence of "partial sums", "partial products" or "partial power towers" tends to a limit as $\ n\to\infty,\ $ then we say the sequence converges as $\ n\to\infty.\ $
See also:
https://en.wikipedia.org/wiki/Tetration#Infinite_heights
