All of the non-analytic smooth functions I've come across so far have either been piecewise-defined or involved infinite sums. Are there any examples of non-analytic smooth functions that consist of a single expression with a finite number of operations?
If the answer is no, what theorem or lemma can be used to prove this?
All elementary functions are analytic on any open set of their domain. Compositions of analytic functions are analytic. Sums, products, and quotients, of analytic functions, are also analytic (as long as the function we divide by is non-zero).
Therefore it is not possible to find a function which is smooth at a point yet non-analytic there, unless you step outside of finite compositions/sums/products/quotients of elementary functions. That is, to get a smooth non-analytic function out of elementary functions, the function which you define has to be defined 'piecewise'.
Of course there are also non-elementary functions.