The bottom bracket spindle on my mountain bike is 111mm. The upgrade I am looking at
(Stronglight Twister Ti) is offered in 113 and 108. Which one should I get and are there any
supplemental pieces I might need to make it fit?
Paul

No way to tell with info given. Check the existing chainline and how far your inner chainwheel is from the frame, as well as how much adjustment is left on the front derailleur. That will tell you which way to go or if neither is satisfactory. Sheldon Brown discusses chainline in detail. The chainline can be adjusted to the right with a spacer but you need to look at where that puts your left crank, how far in the left cup needs to go and if you are losing significant thread engagement on the right.

For gram counting, a Phil Ti / Mg BB will dial in better, as It offers many spindle lengths
+ the mounting rings can be shifted in the frame a few mm
and the press fit spindle , can also be moved in the BB cartridge.