My reputation is at this moment at $1600$.
I did some experimenting with $1600$ and obtained the following:
Evidently, it is a perfect square $1600=40^2$
Also, it is a hypothenuse of a Pythagorean integer-triple triangle $1600=40^2=32^2+24^2$.
Also, it can be written as the sum of four non-zero squares $1600=20^2+20^2+20^2+20^2$
So, $1600$ is a perfect square, a hypothenuse of a Pythagorean integer-triple (so can be written as a sum of two non-zero squares), and a sum of four non-zero squares.
Is there an infinite number of numbers like $1600$? Can you find some more?
Edit 1: It is also a sum of $4$ non-zero positive cubes: $1600=8^3+8^3+8^3+4^3$
Edit 2: It is also a sum of powers from $1$ to $4$, as we see $1600=7^1+9^2+6^3+6^4$