if you like racing then F355 is your thing, one of the best simulations ever, can't say it's pure arcade fun (because it isn't) but it's one of the most rewarding ones
on the arcade side we have shutoku battle 2 where you need to battle opponents to win money and upgrade your cars, it's tough but very fun and intriguing, or sega rally two which is pure fun. Don't know what others think but i like buggy heat very much, with good graphic and cheaper to find on the web.
if you like sports then are virtua striker which, even if not really a pure soccer game, it's one of the greatest (even if it can be frustrating at times), j-league spectacle soccer is another great game with very good graphic and good gameplay, is definitely the soccer game for the system. Then there are other sports, with virtua tennis, the 2k series which honestly beats ea hands down and if you like golf, we have netto golf which is one of the best golf game ever...
if you like fighting games you need to check guilty gear, a sprite based fighter, project justice which is one of capcom best games and heavy metal geomatrix which is very fun to play
if you like beat em ups you should check cannon spike and dynamite cop, they are a blast to play and in co-op the blast is even doubled
on the adventure side we're covered with shenmue which is the greatest and most innovative game ever (i dare others to contradict me :p ) and the resident evil series which is still a masterpiece nowadays
of course there are plenty of other games but those i mentioned will be a good start...