Last digits of power towers $7$, $7^7$, $7^{7^7}$, $7^{7^{7^7}}$, ... don't change, and generalisation While playing around with Wolfram Alpha, I noticed that the last four digits of 
$7^{7^{7^{7^7}}}, 7^{7^{7^{7^{7^7}}}},$ and $7^{7^{7^{7^{7^{7^7}}}}}$were all $2343$. In fact, the number of sevens did not seem to matter (so long as it exceeded five); the result was always $2343$. With further explanation, the last five digits appeared to remain unchanging with at least six sevens, and so on. When I replaced the sevens with other digits, the same effect appeared (even for digits not relatively prime to $10$, which somewhat surprised me). 
My question is: can someone shed some light on why this is taking place? I have some [extremely] basic knowledge of [extremely] elementary number theory, and I've noticed that $7^{2343} \equiv 2343 \pmod{10^4}$ and similar results for other exponents and quantities of exponents, which explains why the pattern continues once it exists, but I haven't had much luck in proving that this will always work. I greatly appreciate any help anyone can provide me for explaining this interesting result. Thank you in advance.
Edit: I recently found that this question is strongly related to Problem 3 of the 1991 USAMO, the solution for which can be found here.
 A: This isn't a complete answer, but perhaps something that might be part of one. 
Consider the equation $2^x \equiv x \pmod{100}$. The smallest positive solution is $x=36$. It also turns out that $36^{36} \equiv 36 \pmod{100}$. Thus if we have $2^{{36}^k} \pmod{100}$ it will reduce to $2^{36 * 36 * \dots * 36} \equiv (2^{36})^{36 * \dots * 36} \equiv 36^{36 * \dots * 36} \equiv 36 \pmod{100}$, so the residue will remain fixed no matter what power of 36 we raise 2 to. Something similar might be happening with your powers of 7 modulo 10000. 2343 is the smallest positive solution to $7^a \equiv a \pmod{10000}$. If we swap the roles it's also true that 7 is the smallest positive solution to $2343^a \equiv a \pmod{10000}$. I haven't thought about this long enough to know whether I should be surprised.
Let $(a,x)$ denote that $x$ is the smallest positive solution to $a^x \equiv x \pmod{10^d}$, where $d$ is the number of digits of $x$. I found the pairs (2,36), (3,87), (4,96), (5,25), (6,56), (7,43) [which contains the last two digits of your 2343 value], (8,56), and (9,89). Solutions to $a^a \equiv a \pmod{10^d}$, where $d$ is the number of digits of $a$, can be found here. The sequence itself is pretty interesting.
