I would not exactly call myself a completionist, since I generally do not bother myself with things such as online achievements and whatnot.
That said, I do tend to be a completionist in games where completing additional tasks actually nets you certain in-game benefits. For example, in RPGs, I like to do all the side-quests that I can in order to delve deeper into the story as well as gain additional experience.
An example of games which I personally think reward completionists very well is the Super Robot Wars franchise:
- First, there is the fact that if you want to play the additional missions and experience the REAL ending, you have to collect a certain amount of Battle Masteries. These are optional objectives each mission has that make that mission more challenging (by, for example, providing additional tasks and/or limiting the amount of turns in which you can complete the mission).
- Second are the secrets. These are basically hidden items and mechs which you can obtain by completing certain steps throughout the game. However, the games generally tend to have branching paths at several points, making it impossible to get all the secrets in one playthrough. So in order to unlock all the secrets, the player must play the game again.
- Third are the additional modes. Basically, once you complete the game the first time around, you unlock an additional mode in which you get to play the game with the additional skill points and money you earned through the first playthrough. And once you complete that, you unlock another mode which is the same as the first playthrough, except you also have one of each mecha available to you right from the beginning.
Overall, if it is just for having a small icon and title on the internet, I would generally not bother. However, some games (like the aformentioned SRW) do make being a completionist very worthwhile.