I have owned a Dreamcast, a Gamecube, an XBox and (fairly late) a PS2 so I cover lots of bases in there... not that many games I love compared with previous 2 generations though.
I also owned a Game Boy Advanced but I have almost no memories of that...
I regrouped some games of the same series that really feel like variations of the same game, but I put the one I'm voting for first.
1) Jet Set Radio Future
Sega took everything that was so great and fun about the first game but corrected all the drawbacks (controls and camera issues and game length mostly) to make it the coolest and funnest game ever... riding through Tokyo, tagging the streets and making crazy tricks to the sound of the coolest (well, 2nd coolest, original was better) soundtrack make a perfect gaming experience
2) Skies of Arcadia
I actually played Skies of Arcadia Legends, but the differences are minors. That's my 2nd favorite RPG ever, just beside Final Fantasy VII. The "normal" battle systems is nothing special and the story and characters are "only" good but the world (pirates on flying boats !), the ship combats and all the extra things you can unlock (wonders, crew, elite fights, your own island to fill...) gave the game so much depth and fun. Bust most of all it's go some "je ne sais quoi" that makes it specifically endearing.
3) SSX 3
Favorite racing game ever, perfect controls, crazy shortcuts everywhere, characters' coolness turned to 11 and a fantastic soundtrack (that N.E.R.D remix...). The follow up was a bit of a letdown but the recent XBox 360 version was a nice surprise (though multiplayer onlin only ? f*** that !)
4) Time Splitters 2
I also liked Future Perfect a lot but that wasn't much of an improvement. The solo story of Time Splitters 2 is the funniest and the challenges to unlock stuff were time consuming, but that would also rank very high in my favorite multiplayer games, taking everything you liked from Goldeneye and Future Perfect and refining it, with better bots AI, more game modes and one of the only level editors I actually used.
5) Guitar Hero 2 (Guitar Hero series from 1 to 3)
Why 2 ? Well first pull-ons and hammer-offs that were almost impossible to do in the 1st game became a 2nd nature while playing this one. But beside this technical point the game was longer, looked better, had cooler characters. And to top it off that's the one with the best tracklist (until more recent games where you can download more, you can't beat Rock Band 2 on that aspect) : specifically going through Free Bird's solo, Sweet Child o'mine intro (with pull ons and hammer offs only once you got used to it), blazing through Beast and the Harlot (tough one... though Psychobilly Freakout was the worst) or just unleashing the raw power of Search and Destroy. Some of the best moments are on the follow up (My Name is Jonas !) or the 1st game (Ziggy Stardust) but the 2nd had it all.
I beat 1 and 2 (and I think 3 but I have a doubt about Cult of Personality) on the hardest mode, I got to a point where it was all muscle memory so I was actually better if I didn't think I was playing a game... that's something I first noticed playing Bust a Move 2 where I would be better if I was talking with someone as playing than if I really tried to focus... that's a strange feeling but also a great one.
6) Jet Set Radio
As I said, the game had some defaults that actually hit me hard when replaying it recently on xbox 360. But still, it was so fun and endearing that it will always be one of my best video games memory ever. Plus best soundtrack ever.
7) Psychonauts
8) Fable
Tim Schafer > Peter Molyneux, if only because he seems much more sympathetic. But while Schafer always delivers, Molyneux promises too much. Still both those games are great, original, with cool stories and lots of cool gameplay ideas. Psychonauts wins on its wit and truckload of fun gameplay ideas, nothing has been remotely similar since.
Your decisions in Fable 1 did not impact the world around you as much in the great Fable 2 or the meh Fable 3 but it was still a very cool game. Big issue was that you get too overpowered at some point and the end of the game is too easy, it didn't spoil the game like it happened for Final Fantasy IX for instance, but it still is an issue you too often find in otherwise great game (see Grandia 2 below where I could kill pretty much everyone with Tio before anybody else could act... made the final fights a tad boring).
9) Marvel vs Capcom 2
10) Eternal Darkness: Sanity's Requiem
I never enjoyed the Resident Evil games much so that's survival horror for me... the madness gauge (that you also find on Call of Ctuluh's paper role playing game) is a fantastic idea, breaking the 4th wall with talent : sometimes your character would go crazy and see horrible stuff, but sometimes the game would try to make you feel crazy : turning the volume up and down, disconnecting your controler or showing a blue screen...
The controls were a bit hectic but everything else was great : the story, the characters (it's actually great for a change never to know if the character you play in a chapter will survive or not) and the very accurate for the time depiction of real places : I invite you to visit Notre-Dame of Amiens after playing the game, that's quite striking (plus that's the best cathedral in France, and I'm not saying that because that's where I'm from)
11) Super Smash Bros Melee
Still my favorite iteration of the game, Falco FTW !!!
12) Dynasty Warriors 3/4/5
Dumb ? Yes ! Repetitive ? Yes ! Hours of fun ! Hell yeah !
13) Sega Soccer Slam
My favorite non Jet Set Radio Sega game is not a game for a Sega console, but to be fair Dreamcast is the only one I had. Anyway, 3 on 3 soccer with silly characters punching each other is much more fun that it should be allowed to.
14) Persona 3
I didn't play 4 and didn't have the courage to go that far in the 3, but still, the gameplay, story telling and art design is astonishing and not like much else I have seen.
15) Heroes of Might and Magic IV
That moment when electricity turns off after 4 hours of local arena game on a Heroes of Might and Magic game...
16) Donkey Konga 1 & 2
17) Grandia 2
18) Time Splitters 3 : Future Perfect
19) Ready 2 Rumble Boxing
20) Fire Emblem : Path of Radiance
21) Warcraft III
22) Baten Kaitos
23) Def Jam Vendetta
24) Fire Emblem : The Sacred Stones
25) Rayman 3
26) Civilization III/IV
27) Max Payne
28) Star Wars: Knights Of The Old Republic
29) Tales of Symphonia
30) Kung Fu Chaos
I unfortunately haven't played (or at least not much) : Okami, Beyond Good & Evil, Zelda Wind Waker, Half Life 2, Ico and Shadow of the Colossus.
And I very strongly dislike Super Mario Sunshine, hands down the Super Mario game I like the least.
soundtrack :
I said it before and I'll say it again, Jet Set Radio has the perfect soundtrack, both fitting the game spirit, tempo and design perfectly and able to stand on its own.
Never been much of a dancer, but put Sneakman or Let Mom Sleep and I'll just go crazy moving around like a madman.
One of the best memory still is grinding toward the sunset to the sound of 'Bout the City... some feel good moment (even with the clock ticking as I wonder where the heck the last tagging spot is)
But really, Rock it On, Sweet Sould Brothers, Super Brothers, Magic Girl... so many awesome tracks.
I like the soundtrack of Jet Set Radio Future a bit less, most of the remixes making the songs more futuristic (which went fine with the more futuristic design of the game) but also less enjoyable.
Except maybe "The Concept of Love" :
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Most of my other VG music memories of this era are not original music, whether I was hitting bongos to the sound of the William Tell overture, riding down a peak to The Chemical Brothers' Leave Home or learning to play Bulls on Parade on my plastic guitar without looking at the screen.
But unfortunately, this terrible character selection screen from Marvel vs Capcom 2 is also stuck in my brain...