Turing machines were able to store only one symbol at a time, traditionally -- and it was simply that the tape (the program) was hypothetically infinite. That's my understanding at least.
Also, the data storage isn't infinite on anything :-). --(this comment by Hakware at 16:17, 14 November 2007 UTC; please sign your comments with ~~~~)
- Well, even if real infinite storage is impossible, Turing-completeness still requires theoretical one. In TMs the tape is unbounded, it is finite at given time but can grow infinitely (thus not finite), whenever more of it is needed. But in this, can you make a program that'd grow memory forever? Such as brainfuck's +[>+] ? --Keymaker 21:58, 14 November 2007 (UTC)