The quarter and half farthings were originally meant for Ceylon, while the third farthing was struck for Malta.
The half farthing was made legal tender in Britain in 1842, so you could probably consider that the smallest ever British denomination. But at the time, Malta was considered part of Britain, and so the third farthing might have claim to be the smallest, even though it never circulated in Britain itself.
No-one thought we needed such small denominations, but based on the RPI, a quarter farthing would be worth about 2 pence today, and we still make use of coins worth half that.