That's really tough to answer, because you have the tail end of the PS1, all of the PS2, and the GBA, DS and PSP libraries...
I can rattle off 50+ games that are worth playing, because the first half of the 00s is the golden age of JRPGs imo, but like:
Shadow Hearts and Shadow Hearts: Covenant
Mana Khemia 1 and 2, the Atelier Iris trilogy, Atelier Rorona and Atelier Annie
Wild Arms 3, 4 and 5 (Alter Code F is 'good' too, but I prefer the original)
Suikoden 3 and 5 (and I really like Tactics too!)
Dragon Quest 7, 8 and 9
Fire Emblem 6, 7, 8, 9 and 10
Vagrant Story
Xenosaga 3
Persona 2: EP, Persona 3 and Persona 4
SMT 3, SMT Digital Devil Saga 1 and 2, SMT Devil Survivor
FFX, FFXII, FF: The 4 Heroes of Light, FF Tactics Advance, the FF3 and FF4 remakes
Kingdom Hearts 1, 2 and Chain of Memories
Tales of Eternia, Tales of the Abyss, Tales of Graces, Tales of Vesperia and Tales of Legendia
Front Mission 3 and 4
Grandia 2 and 3
Valkyrie Profile 2
Breath of Fire: Dragon Quarter
Like a billion Disgaea entries (1-3), plus the other SRPGs NIS was making during this period
Eternal Poison
The World Ends With You
Mario and Luigi: Superstar Saga and Bowser's Inside Story
Paper Mario and Paper Mario: The Thousand Year Door
Etrian Odyssey 1 and 2
Pokemon Crystal (plus others)
Golden Sun and Golden Sun: The Lost Age
Mother 3
Summon Night: Swordcraft Story 1 and 2 (I'm not including JP only games, but hgnn, the Summon Night series and the rest of FlightPlan's games during this era are personal favourites)
Mega Man Battle Network 1-6
... I'm gonna stop because the list is frankly stupid already, and that's not 50% of what I'd mention (don't think I mentioned a single PSP game). I know you're going "STUPID STRINGS, I JUST WANT THE BEST ONES!!!" but there is so many stupidly good JRPGs from this period.