It might be more accurate to say the Twitter Fail Whale is infamous rather than famous. It's the whale that shows up when things go wrong on Twitter. Back in the early days of the company, the Fail Whale showed up often enough that it became a bit of an insiders' joke.
People have gotten Fail Whale tattoos, while others have turned it into classically inspired works of art. There's even a Fail Whale Fan Club on Twitter.