I think the 'rarest' coin I have isn't all that rare (6 examples now known).
I feel it's very hard to be certain about 'rarities' in respect of currency coins with so many detector finds nowadays. One coin I have was a new type discovery in 1995, but since publication and listing in Spink a further 13 have turned up, so it didn't stay unique for long!
Patterns and trial pieces might be a bit easier to be certain, especially if it's clear how many have gone to museums and the like.
The rarest 'thing' I have is probably a small Royalist medallion from the 1640s, of which I'm only aware of one other example apart from the British Museum's and which Medallic Illustrations classes as 'Very rare'. But the likelihood there are others, in personal collections (or in museums, poorly or incorrectly identified), has to be pretty high, given these were mass produced items (albeit not in huge numbers when you compare to how many - what are now considered as rare - coins were struck).
Similar to Sword, an example in above average condition (or for me, with decent provenance) is now more attractive to me than simple rarity. I have parted with many rarities, because in the end I didn't find them pleasing and valued being able to buy something commoner, but that I would enjoy owning, more.