Is there any non-trivial algebra for which any non-constant algebraic expression has a root in that algebra?
For example the complex numbers have a solution for any basic polynomial, but do not have a solution to:
$$a (a (a a)) - (a a) (a a) + 2 = 0$$
So we could "extend" the complex numbers with a root to this. But there are likely still other algebraic expressions that don't have a root. So we add a solution to that ... will this process be able to terminate?
For the purposes of this question let's define a "$k$-closed" algebra to mean a root exists for any non-constant algebraic expression represented as a tree of no more than $k$ operations in that algebra. For example, the complex numbers would at least be 3-closed. If there is no non-trivial $\infty$-closed algebra, is there at least a $k$-closed algebra for each finite $k$?