I've actually got quite a few. If you want to try em, pick them up as soon as you can.
1. Threads of Fate
http://www.romulation.net/files/screenshots/roms/PSX/24236/s_0.jpg
One of my favorite RPGs of all time. The graphics, music and characters grasped my attention. Playing as both Rue (a warrior and shapeshifter) and Mint (a magician with a variety of spell types) really made me love RPGs in the first place. If you haven't played it, you will not be disappointed!
2. Banjo-Kazooie and Banjo-Tooie
Possibly the epitome of 3D platforming at its best. The only 3D platformer that could come close IMO was Super Mario Bros. 64 and even then that's a
close second. The controls were slick, the graphics were eye-popping for the time and the characters were really memorable. Play it if you haven't.
3. Ikaruga
http://darkzero.co.uk/asset/2008/04/ikaruga-2.jpg
A truly great arcade-style shooter. How anyone could miss it is a crying shame. Its almost up there with Gradius and Lifeforce. If your a fan of scrolling shooters, I guarantee you give it a shot.
4. Brave Fencer Musashi
http://www.rpgdreamers.com/rpgworld/brave/brave1.jpg
Two words: Frickin' awesome. Combining various sword moves in addition to the ability to absorb the powers of your enemies really mixed up combat and kept everything fresh. The overall presentation is bound to make you wanna finish playing this game. Give it a go.
5. Wild 9
http://loot-ninja.com/wp-content/uploads/2009/01/wild-9-500x373.jpg
I was shocked to find that almost no one remembered this game. The first time I played it was on one of those old PS1 game demo discs that Pizza Hut gave away as a promotion. Needless to say, I went and bought the full version and I can say it was pretty good. Not the best game I played but definitely a memorable experience.
6. Xmen: Mutant Academy 2
http://image.gamespotcdn.net/gamespot/images/2001/ps/xmenmutantacademy2/xmenmutantacademy2_790screen001.jpg
Another game I bought because of its demo, it really is a very deep fighting game that really makes you want to actively seek out people to show it to. This is what Xmen vs. Street Fighter should have been like, with tighter controls and a better presentation.
7. Oddworld: Abe's Odyssey
http://laputanmachine.files.wordpress.com/2010/04/oddworld_abes_oddysee.jpg
People seem to remember Abe's Exodus more than this one. This game is more about trial and error rather than just jumping in. Each and every situation requires a bit of problem solving and thinking outside the box to get around that will really make you feel like accomplishing something. A good platformer in of itself.
8. SSX3
http://www.nintendospin.com/wp-content/uploads/2008/11/ssx-3-image1.jpg
The game that made me almost forget about most other snowboarding games. With all the over-the-top moves as well as the many different venues, characters and customization sucked up hours of my time for that game and I can it was time well spent.
9. Soul Blade
http://www.romulation.net/files/screenshots/roms/PSX/24860/s13d9e79848827958b117ade48a627330.jpg
The game that actually started the Soul Calibur series.
So those are the four I could pick, mainly since there are so many I could choose from, but this list is getting way too long.