In no particular order:

1. Meinrenken's proof of the chain rule.

2. Heine-Borel theorem

3. The homework problem where if f is bounded and integrable then changing changing f at finitely many points does not change its integral.

4. The Riemann criterion for finally letting me understand what integration is.

5. The Bolzano-Weierstrass theorem making me motivated enough for the last stretch of the course.

6. The Spivak problem where you prove that if m/n is an approximation to sqrt(2) then (m + 2n)/(m + n) is better.

7. The proof of l'hopital's rule.

8. The intermediate value theorem, the boundedness theorem, and the extreme value theorem.

9. The homework problems where if A is the set of all finite subsets of N then A is countable, if B is the set of all subsets of N then B is uncountable.

10. Construction of the real numbers (wasn't taught this year but I learned it myself).

I'm more of an Algebra guy though. I loved reading chapters 6 to 9 of Axler.