Any car is eligible for EITHER the Turbo or NA events -- some can enter both.
Any car is eligible for ONE of the FF, FR, MR, 4WD events.
Any car is eligible for EITHER the Race Car events (just Formula GT) or Normal events (PD Cup, Trial Mountain Enduro)
Many cars are eligible for the 8 non-wet rally events.
So, what it really comes down to is finding a car that can rally, switch between NA and Turbo, and that is also able to enter the most of the following:
1 Lightweight Cup
2 Stars & Stripes
2 Spider & Roadster
2 Boxer Spirit
2 80's Cup
2 Tourist Trophy
1 Italian Avant Garde
2 Race of Red Emblem
2 Legend of Silver Arrow
2 Altezza Race
2 Vitz Race
2 Type-R Meeting
2 Evolution Meeting
1 Beetle Cup
1 Elise Trophy
1 Clio Trophy
1 Tuscan Challenge
I can't think of a car that can enter more than 4 races (2 events) in that list. The Miata can do 80's cup, Spider & Roadster and Roadster Endurance, and it can do both NA/Turbo, but it can't rally. The Skyline R32 can rally, it can enter Red Emblem and 80's, but it can't enter NA.
That means that the car that can enter the most events is the Acura Integra, because it can enter both Stars & Stripes, both Type-R Meetings, NA and Turbo events, Rally events, and all the standard stuff.
Acura Integra Type R: 50 events
Beginner: 8
Sunday Cup
Clubman Cup
FF Challenge
Stars & Stripes
Race of NA Sports
Race of Turbo Sports
Type-R Meeting
GT World Championship
Amateur: 13
Japanese Championship
American Championship
European Championship
GT World Championship
German Touring Car Championship
FF Challenge
Stars & Stripes
Race of NA Sports
Race of Turbo Sports
GT All Stars
All Japan GT Championship
Type-R Meeting
Dream Car Championship
Professional: 10
British GT Cup
GT World Championship
FF Challenge
Race of NA Sports
Race of Turbo Sports
GT All Stars
All JGTC
Dream Car Championship
Polyphony Digital Cup
Like the Wind
Rally: All 10
Endurance: 9 - All except Roadster Endurance