Here are just a handful:
My favorite Twitterbot is CBBOT. CBBOT grabs data from Crunchbase (Crunchbase contains a database of information about Web 2.0 start-ups) using Crunchbase's API - Then lets you search that data semantically using Twitter. So for example, to quickly find out who founded Twitter, just ask Twitterbot account @CBBOT "founder of Twitter" would return info. about Twitter and it's founder, Biz Stone.
@Tipr lets you send restaurant bills, and it will tell you how much to tip
@ Wordbot lets you send a string of scrabble letters (ZJFDIRFA), and it will return a list of words you can make.
I expect to see many Twitterbots as business finally learns what they can do... How about sending a Tweet to flighttracker.com "American #3432", and getting updates about your flight...Shopzilla "Nipon Coolpix" and getting best prices for that cool camera your looking to buy. Your a business, why not create a bot to let customers mine your FAQ and knowledge base?
If you have ever used IM bots, expect to see all of them ported to the Twitter OS, in the mean time, there is a pretty good list of bots here .
Probably the most ambitious app yet is Phweet. Phweet lets you make VOIP calls over the Twitter network. The best example of this was when someone successfully made a call using Phweet from an airplane.
Want to paypal someone over the Twitter network, try Tipit.
New and cool is Mozilla's Ubiquity ... Ubiquity is much more then just Twitter, but in a nutshell Ubiquity 0.1 will let users perform such tasks as e-mailing friends or colleagues maps of locations; translating text to a different language directly from a Web page; searching key sites including Google, Wikipedia, Yahoo, YouTube, Amazon.com, Digg and Twitter; and finding and inserting reviews from local search site Yelp.
Another new (to me) one is Twiffid. Twiffid lets you (via Twitter) see what you Twitter friends are blogging about. Know of other good one's? Send me a Tweet
An important one (if you blog) is Twitterfeed. Every time you update your blog, Twitterfeed will automatically post to Twitter letting all your followers learn immediately of your new blog post.
Want the BIG list? Over 140 apps. are listed here