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.

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".