Degrees are also sometimes associated with kyu ranks, mostly at brown belt. Some systems may call a 3rd kyu brown belt a "3rd degree brown belt," a 2nd kyu brown belt a "2nd degree brown belt" and a 1st kyu brown belt a "1st degree brown belt." Other systems might reverse that.
Sheslusa is correct, that different styles will have different limits on what the highest rank within the style can be. Most use the 10th Dan rank as the highest rank possible. Some only go up to 5th Dan, while certain systems, as suspected, go all the way to 15th Dan. It can and does vary, even within a given system, since there might not be any 10th Dan holders still alive.
Just as a GENERAL guideline, regarding Dan ranks, most systems will have rigid standards regarding what is needed to go up to your 4th - 5th Dan rank (assuming we're talking about a 10th Dan max system). After that, promotions are generally given out based on what that person has done for the style, and his accomplishments. This is where it can sometimes get ugly in a political way. Not too common, but it does happen.