Since I can list nearly two dozen off the top of my head, I started wondering how many different RPGs there are. Is there a list anywhere where every published RPG is listed?
RPGGeek is a very exhaustive RPG database (including related things). As of February 2013, it lists 29541 individual published RPG items (that is, books, fanzines, published PDFs and so on) belonging to, according to their count, 4007 different RPGs.
You can browse the list of all “individual RPGs.”
These are published RPGs (all homebrew is grouped under one "misc" individual RPG item), including LARPs. It includes free games (for example, Game Chef 2012: Last Chance is an entry listing all the contestants for the 2012 Game Chef competition, which are all small free RPGs made during the one week long competition period), and significantly varying different editions are counted separately (cf. Call of Cthulhu 1st Edition vs. Call of Cthulhu 2nd–6th edition, which are two separate RPGs on that list at the time of writing this).