Last game is usually going to be some dancing/singing/plastic guitar game or football (as in soccer) game, usually one aimed at developing countries and the few holdovers in the older set of first world that buy it anyway.
One of those is most likely to be exclusive as xbox and gamecube (I assume we are not counting PC in this, not that the PC would necessarily have got those either) would have probably trailed off a year or two before then. That or PAL countries might get a very late stage version of a RPG that was previously Japan/NTSC only.
Your best bet to find out, or find out the last interesting game if sports game/music game is not your thing, is most likely to be find a Scene release list you can narrow to PS2 games (sadly the old ABGX site did not follow PS2 back when despite being most noted for xbox
https://web.archive.org/web/20190906021919/http://www.abgx.net/ https://web.archive.org/web/20190902144503/http://www.abgx.net/xbox_releases_alpha.txt ). I also don't have much in the way of good Scene release/lister sites right now, much less that go back that far and that you can reasonably program so you might have to go searching there (most releases should note PS2 or something like it in the name*, barring the useless PS3 plop the disc in and copy rips then PS3 stuff would not be along for a while, if it is bundled with console games in general then have fun).
*
https://web.archive.org/web/20160930230045/https://scenerules.org/p.html?id=2008_CONSOLE.nfo (
https://web.archive.org/web/20160930224951/https://scenerules.org/ for the sake of a more general link) mentions there should be PS2DVD/PS2CD in the name but I don't know if there was any kind of split as I was not really following the PS2 back when, not to mention the earlier stuff that might have dodged it.
You might have to filter out any redumps (PS2 was not quite the horror story of the PS1 scene but it was hardly as "
professional" and systematic as the 360 and stuff since has become) and possibly region oddities or alt versions, though region oddities and alt versions could also plausibly be that.