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[Immortal]

A few things to try.
Recursive tree pictures! Make them grow over time and sprout fruit.

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Make board games! Corridors, abalone, chess are all very fun ones to make, each with a different skill needed to program them.
3D Rotation.
Create a triple linked list! You'll feel ridiculously satisfied if you can get it to balance. Even if you only get a few nodes to rotate.

[Starver]

If you don't mind it being a simple and unoriginal idea:

A program where "how many monkeys, indeed, does it take to eventually write out Hamlet... or maybe just the first page of it" is asserted.
Not by truely randomly doing 100 random threads of totally random characters until one gets it right... but some other way. Maybe ... giving the chances for a monkey to type the correct character a boost each time one monkey gets it.

Also, kill monkey after each mistake.
Maybe add the ability for text files to be read, or strings to be typed.
((Was an idea bouncing around a long time ago that I just fished out. I might do this sometime soon XD
Sadly, it's not original...Probably dozens of these programs sitting in unused and unloved sectors of old harddrives globally.))

Yup, done some of this myself...  Maybe browsing this site and its messageboards (although long since shut down due to spamming threats) might dig up some of what I did, or generally inspire something else along those lines, which I'm sure no-one would mind being taken up.  (No, I'm not the site's author/owner, but I was one of the regular board contributors, back in the day, and have developed some of the ideas for my own fun-type use, elsewhere...)

[malloc]

A suggestion to put forward:

Write a program to draw the Maldelbrot set (or another fractal of your choice) such that you put in the co-ordinates or the area to plot, the number of pixels to use, and it generates it. I did one of those earlier today and it was quite a nice experience. If you're feeling artistic, maybe have the different pixels colored according to their place in the fractal?

I can agree. This is actually a pretty fun thing to do, also, it's like the hello world of graphics programming.
But honestly, it takes like half a minute to write a shader that can produce a Julia or Mandlebrot shader nowadays, so it's not much of challenge.

But it's weirdly satisfying to do. Especially the first time.

[Kofthefens]

Sorry I haven't done any since the first. I've got a lot going on right now, so I probably won't be able to do more for a while. Thanks for all the suggestions though, hopefully someone else will find them useful.

[alfie275]

Something todo with artificial life. Perhaps a simple grid based world with genetic algorithms?

[Virex]

Try your hand at the following (pdf link) http://www-users.cs.york.ac.uk/kazakov/papers/aamas-paper.pdf. see if you get something interesting.

*cough cough*