In full generality, there provably isn't any method for complete simplification (i.e., bringing an expression into a canonical simplest form).  Simplifying should have two key properties: it should be algorithmic, and simplifying two different expressions for the same thing should give the same simplified form.  If you have a simplification method with these properties, then it gives an algorithm for deciding whether two expressions are equivalent.  However, Richardson proved that there is no algorithm to decide whether two closed-form expressions define the same function.  (Of course you have to specify what you consider "closed-form".  See D. Richardson, *Some Undecidable Problems Involving Elementary Functions of a Real Variable*, Journal of Symbolic Logic **33** (1968), 514-520, http://www.jstor.org/stable/2271358.)

Of course simplifying becomes easy if you give up on these properties.  If you don't care about algorithms, just choose a representative for each equivalence class arbitrarily and declare it simplified.  If you don't care whether equivalent expressions simplify to the same result, then just declare everything is already simplified.

This argument rules out only a very general notion of simplification.  It still makes sense in many important special cases, and as Joel David Hamkins observes in the comments, one could still define a notion of simplicity even if there is no full simplification method.