If you have some patience, I will bet on Merlin engines, flown on SpaceX Falcon 9 and Falcon Heavy launchers. They use 10 per launch. 9 on the first stage, one on the upper stage.
In only the first 8 launches (as I write this in early 2014) that is 80 engines. With 50 flights on their manifest to clear, they will hit several hundred engines flown in the next year or two.
Updating this at the end of 2019, there were 76 Falcon 9 flights (760 engine flights), and 3 Falcon Heavy with 28 per flight (84), ignoring the Falcon 1's 5 engines, that is 844 with 20+ flights planned for 2002. That exceeds the SSME and SRB combined numbers already.
Burn time would be fun to compare:
- SSME burn time, 480 s (405 burns)
- SRB burn time, 124 s (270 burns)
Total: 227,880s, 63 hours, 18 min
- First stage Merlin burn time, 158 s (765 burns)
- Second stage Merlin burn time, 397 s (79 burns)
Total: 152,233s, 42 hours, 17 min.
But with 75,000 seconds difference, and about 1819s per mission, Merlin only needs 41 more flights to catch up.
Of course first and second stage burn times vary somewhat based on mission flight profile, this is meant only as an approximation.
However, as of the current time, the Soyuz booster with the RD-107 is the clear winner with 700+ launches.
SSME with 135 launches X 3 engines per launch is 405 full duration firings (Maybe -3 for Challengers last flight). Then of course the SRB's with 270 full duration firings on those same launches. Now, are the SSME's still in use? Well not on the Space Shuttle, but in theory on the SLS now known as RS-25D or RS-25E engines.