Question: User: 2^^5
1. The problem is to evaluate the expression $$2^^5$$, which represents tetration, a power tower of 2's of height 5.
2. Tetration is defined as repeated exponentiation. For example, $$2^^3 = 2^{2^2}$$.
3. We calculate from the top down:
$$2^^5 = 2^{2^{2^{2^2}}}$$
4. Calculate the top exponent first:
$$2^2 = 4$$
5. Next exponent:
$$2^4 = 16$$
6. Next exponent:
$$2^{16} = 65536$$
7. Finally:
$$2^{65536}$$
8. This is an extremely large number, much larger than can be written out explicitly.
9. So the value of $$2^^5$$ is $$2^{65536}$$, a power of 2 with exponent 65536.
This completes the evaluation of $$2^^5$$.