The slimmest fountain pen kit on the market is the Beaufort Mistral to the best of my knowledge, and it was deliberately designed to win that accolade. The fittings on the nib barrel are 12.25mm at their widest - that constraint being necessary because of the need, as Mark mentioned above, to get a cartridge or a converter in and out of it. The cap is wider than that of course, because it has to screw on to the nib barrel.
The thinnest cap on a fountain pen is the Traditional - that's because the cap pushes on instead of screws on. Berea introduced their New Series fountain pen a while ago, which is marketed as slim, but it relies on cutting away some blank to expose a tenon, in the same way as a Round-top or Flat-top Euro BP. In my opinion the new series is both ugly and flakey (and mis-named, and not very thin).