1. Indiana Jones
2. Bourne
3. Lords of the Rings
4. Star Wars
5. Matrix
6. Pirates of the Caribbean (not really a trilogy now, but oh well)
7. Die Hard
8. Jurassic Park
9. Mad Max
10. El Mariachi Trilogy
11. The Man With No Name trilogy
12. The Mummy
13. Transporter
14. Oceans Trilogy
Hmm that's all I can think of off the top of my head, and no those aren't in any particular order. Also, some aren't actually trilogies since they've had more than 3 made (or about to). And if you really go strict with the Trilogy definition then most aren't trilogies (trilogy being 3 movies that were intended from the start to be 3 movies covering the same story arc).
I can't think of 5 let alone 10. The Bourne films would be number one though
10 is somewhat of a stretch.
Here's a list of strictly favorites:
1. Man With No Name Trilogy
2. Indiana Jones
3. Terminator
4. Mad Max
5. Star Wars
6. The Matrix (To be honest, I haven't seen the 3rd, but I still feel like giving it a mention regardless)
7. Jurassic Park
8. Bourne
9. Pirates of the Caribbean
10. Die Hard
... and if anyone's having trouble thinking of any, here are some more unmentioned:
Batman (if Burton and Shumacher's films count)
Blade
Infernal Affairs (sequels may qualify it as action)
Mission Impossible
Rambo
Robocop
Rush Hour
Spiderman
Underworld
X-Men
I know that plenty of those had more than three films made, but it's difficult to think of much without letting go of that rule.
Problem with trilogies is that they inevitably become quadrilogies, so this may be out of date pretty quick, also, not many good action trilogies left (those that were are all quad or quintrilogies), but...
Spider-man Trilogy
X Men Trilogy
Predator Trilogy
Transporter Trilogy
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Reverend Arthur Belling was to change our whole way of life, and every Sunday we'd hurry along to St Loony up the Cream Bun and Jam.