I've run across a few interesting puzzles on the internet over the years.
The first one I was introduced to (by my brother sam) is notpr0n which bills itself as
The hardest riddle available on the internet. It's an interesting puzzle requiring a reasonable amount of outside knowledge in order to progress. The website says it has 140 levels (of which I think I've seen 30, but in my most recent run - which is still ongoing - I've only hit 10) and has been running since August 2004, making it pretty impressive that you can still go play it.
Another that I've enjoyed is Funny Farm which is an idea connection puzzle. It starts with a mostly-empty tile with one square filled in -
Farm Animals, the category for the tile - and you have to add related things together. Across the puzzle you'll encounter many interesting categories and connections, again requiring quite a bit of outside knowledge in order to complete. You may know farm animals, and some other things those animals are also related to, but can you connect them all the way out to various pop culture and literary ideas? I've never finished this one either, but it's always fun to revisit.
On the more techie side, there's quite a few continuously running hacking/cracking sites out there. A centralized directory and scorekeeper called WeChall is available which helps you find more of this style of puzzle to play. Solving these types of challenges is a bit more practically useful than the other two since they directly translate to work in computer security. These are the same types of problems you will find in capture the flag competitions, which are time-boxed competitions built from these problems. There's a central directory and scoreboard available for CTFs called CTFTime.
Another site with tech-style puzzles is Project Euler. Initially this looks like a set of difficult programming puzzles (since you'll require a computer to solve most problems) but it becomes obvious that you'll need significant mathematical know-how as well. This is another one that'll require some significant time to complete - as of when I'm writing this, there are 608 problems. The most-solved problem (Multiples of 3 and 5, problem 1), has 772508 solvers, but the least-solved (Lattice points in lattice cubes, problem 5579) has only 72.
I'm sure there are many others, but these are the ones I personally have experience with. Let me know if you have spent time on other ones, I'm interested to see more.
On this idea - puzzles in the style of notpr0n seem to be very free-form. I'm curious if there are any of these that have a physical component to them, requiring you to travel to a particular location and find something. I've done letterboxing in the past, in which you are given a set of orienteering instructions (possibly in riddle form) at the end of which is a notebook with the signatures and/or stamps of those who've made the trek before you. It's a fun pastime. An internet-based puzzle that incorporates physical elements like that would be fascinating.
I'm also curious if you could make a puzzle in which when you finish you create another level, building on the puzzle as it was and making it better as time goes by. This might be workable, especially if the puzzle was initially difficult enough that those who finish are highly dedicated - and therefore likely to create good follow-on levels. I'd like to try something like that in the future.
That's what has been on my mind this week. Let me know if you try any of these out.