If you compute it, you will be guilty of:
Copyright infringement (of all books, all short stories, all newspapers, all magazines, all web sites, all music, all movies, and all software, including the complete Windows source code)
Possession of child pornography
Espionage (unauthorized possession of top secret information)
Possession of DVD-cracking software
Possession of threats to the President
Possession of everyone's SSN, everyone's credit card numbers, everyone's PIN numbers, everyone's unlisted phone numbers, and everyone's passwords
Defaming Islam. Not technically illegal, but you'll have to go into hiding along with Salman Rushdie.
Defaming Scientology. Which IS illegal -- just ask Keith Henson.
Also, your computer will contain all of the nastiest known computer viruses. In fact, all of the nastiest POSSIBLE computer viruses.
Some of the files on my PC are intensely personal, and I for one don't want you snooping through a copy of them.
You might get away with computing just a few digits, but why risk it? There's no telling how far into Pi you can go without finding the secret documents about the JFK assassination, a photograph of your neighbor's six year old daughter doing the nasty with the family dog, or a complete copy of the not-yet-released Pearl Harbor movie. So just don't do it.
The same warning applies to e, the square root of 2, Euler's constant, Phi, the cosine of any non-zero algebraic number, and the vast majority of all other real numbers.
There's a reason why these numbers are always computed and shown in decimal, after all.