I'm taking a crack at finding a non-Lychrel number that goes palindromic in more than 261 iterations. See https://en.wikipedia.org/wiki/Lychrel_number .
As of August 31st 2015, the longest known chain is 261 iterations, beginning with the number 1186060307891929990 . This record is from 2005 (!!!), so I figure I have a shot at cracking it.
I'm going to code up something in an arbitrary-precision integer C++ library, optimize it, and try to get some supercomputer time. Hopefully the last 10 years of Moore's Law make up for my lack of algorithmic prowess.