I hesitate to submit this in the company of the more informed answers already present, but it seems that in the special case of polynomials, "simplified" form is basically conjunctive normal form or disjunctive normal form: an outer layer of + or * with all operands being only * or +, respectively. It also seems to me that this pattern is generally accepted as, if not the, then at least a simplified form for any kind of combination of functions. So, for example, $$e^t(1 + t) + t(1 + e^t) = e^t + 2t e^t + t$$ can be said to be simplified on account of being fully disjunctive-normal, having a layer of + followed by a layer of * followed by one of exponentials. By grouping terms one can obtain a narrower tree structure (such as $e^t(1 + 2t) + t$) but some of the deeper nodes (groups) would have operations (namely, addition) that occurred higher on the tree. Many of the objections made in above answers/comments touch on why this kind of tradeoff is inevitable.
Edit: It occurred to me that the statement about "any kind of combination of functions" has more meaning when it's given more context. So, for example, an expression similar to the above can be written in several ways: $$t + (1 + 2t) e^t + (t^2 - t - 1) e^{2t} = t + e^t + 2te^t + t^2 e^{2t} - t e^{2t} - e^{2t} = (1 + e^t - e^{2t}) + (2e^t - e^{2t}) t + e^{2t} t^2$$ in which the first is simplified as an element of $\mathbb{R}[t][e^t]$, the second as one of $\mathbb{R}[t,e^t]$, and the third as one of $\mathbb{R}[e^t,t]$. Some people might even quibble with the order of terms in a polynomial (such as $1 - t^2 + t$ versus $1 + t - t^2$ or $-t^2 + t + 1$), which suggests to me that the whole simplified-polynomial business (as seen in practice if not principle) is a combination of choosing a presentation of the polynomial ring, and choosing a term order as in Groebner bases.
Nonetheless, even if there is no precise definition of "simplify", it is possible to assign a number of criteria, such as the above, that, although being impossible to meet simultaneously, can individually indicate any number of expressions as being "not simplified".