Identifying my Shimano groupset - and upgrading cassette.
My triathlon bike - bought 2nd hand about 10 years ago for £1,200 - has a Shimano Dura-Ace group-set. The cassette has 9 cogs: smallest 12, biggest 23.
I'm trying to find out if I can upgrade to a 10 cog cassette will keeping the same rear derailleur and brake/shifters. I think it's a 7700 group-set.
No, if you want to keep your current shifters, and go 10 speed, it's not possible; as the rear shifter doesn't have enough clicks / positions. You will probably find it easier / cheaper to get a new bike if you want 10 speed.
You can keep your rear derailleur since 9-speed Dura Ace is compatible with other and newer groups. You will need new shifters as jimc101 explained and, unfortunately, brifters are the most expensive single item. You could save a fair bit of money by using 105 or Ultregra brifters unless you insist on Dura Ace.
Collector of Useless Info
If it's a tri-bike, it probably has bar-end shifters. Bar-end shifters usually have a friction mode, but running 10-speed friction mode takes a very steady hand to align the rear derailleur well enough on every shift to keep it from ghost-shifting. In practice, friction shifting is for 8 speeds or less. You'll need new 10-speed shifters, but the derailleurs are probably OK.