Lets recall Platonic construction in plane geometry. It is impossible to square a circle using only ruler and callipers. But is also known that it is possible to do it with ruler which has a mark on it.

So we know by Gaois theory that we can't write expressions for the solutions to all algebraic equations using the four arithmetic operations and nth roots.

Question: is it possible to write expressions for the solutions to all algebraic equations using the four arithmetic operations and nth roots and something else in finite numbers of factors? For example adding logarithm to the set of operations? ( or other function, or class of functions...)?

If Yes: Do they form any kind of algebraic structure?