TITLE: Ugly solutions to easily stated problems
QUESTION [5 upvotes]: I recently saw a very hideous closed form for a quartic equation here:
Does a closed form solution exist for $x$?
For fun, I'm wondering about surprisingly ugly solutions/ complicated machinery needed to problems that are simply stated. 
Clearly, they all don't have to be algebra-based.
I'm trying to get a sense for how particular different mathematic methods are necessary, and interesting. Even for easily stated problems.

REPLY [3 votes]: Classify pairs of commuting matrices up to a change of basis.
This sounds like an extension of Jordan normal form, since the matrices should be simultaneously Jordan-izable or something.
In fact it is the canonical example of a "wild" linear algebra problem that is as hard as classifying $k$-tuples of (noncommuting) matrices for all $k$.  No reasonable solution is expected to exist.