Charged battery, but doesn't even try to start.

I have experienced a similar issue where I get absolutely no response from the battery knowing that it's fully charged. The cables didn't look too dirty but I pulled them off, cleaned everything up really good with a terminal cleaner and that solved my problem. If it's this then it's easy and it's free, if that doesn't do it for you then follow the advice of everyone else as you dig further into the system.