A fair amount of research has gone into this question. A number of very simple Turing-complete languages are known.
valarautical mentioned x86 mov. Another is an assembly-ish language with two instructions: (1) decrement and (2) branch if <= 0. Another is the Lambda Calculus. The only thing it has is lambda (and variables and function application, if you want to consider those to be things). Lisp with only QUOTE, ATOM, EQ, CAR, CDR, CONS, and COND is another example (and perhaps even that list is not minimal; I'm a little vague on this).
