I only play solo and (mostly) improvised games. I'm not saying I would never play with a group but it just did not happen since I started playing again in 2000. I haven't seen anybody from my old gaming group of the 80s since a least a decade. So I started over with a friend and for both of us it's been the most memorable RPG experience. We've started a campaign in Celestia and the Realms months ago and we're still enjoying it. My friend has a his character and I play usually one NPC. So far, battles have been pretty dangerous but I do allow him to max his player a bit, because I do not have the time to adapt published adventures for two players.
Also, we started taking turns DMing, so we play more than one campaign simultaneously (DnD, Ars Magica, maybe D20 Modern). Since there's only the two of us, it's easy to decide what kind of campaign we would like to play, to organize our schedule, to switch between campaigns.
Also, all our campaigns are improvised. We simply do not have the time to prepare games in advance. For my part, I have the general ideas in my head, but I do not prepare encounters and such. So we let the campaign evolve (as proposed by Darklone) and the player participates to the development of the storyline sometimes as much as the DM. This is very easy to do in solo improvised games, since you do not have to put up with a committee everytime you have a blank or you do not know exaclty where the storyline should go. Plus, solo improvised gaming is as much surprises to the DM as to the player.
As for the the type of game you should play, I don't think it really matters. Most published adventures we've played were drastically modified by the end to suit our playing style and the storyline of the campaign, but we did play classic adventures (i.e. Forge of Fury) and we had tremendous fun. For example, in FoF we were badly wounded during a first encounter with the orcs during our break-in. So we had to flee down in the forge and hide. We could not go out nor go deeper. So we hid for weeks down there trying to avoid orc scouting parties, troglodytes, etc... This would have been impossible with a group of four or more PCs. Luckily my NPC had some cleric levels (multiclassing is essential in solo games) so we had food, water, and healing. This is what we still remember the most vividly from our venture in the FoF (alongside the fight with the dragon). So it is possible to slightly modify classic door-monster-treasure dungeon crawls for the needs of solo adventuring and still have a great time.
G.G.