Artin's theorem on positive polynomials, which solves Hilbert's 17th problem in the affirmative, apparently still has no algebraic proof.

Theorem (Artin): If $f \in \mathbb{R}[X_1,\dots,X_n]$ is pointwise nonnegative, then it is a sum of squares in $\mathbb{R}(X_1,\dots,X_n)$.

Artin's proof uses quantifier elimination for real closed fields from model theory, so this arguably qualifies as not purely algebraic. As far as I know this is still the only successful approach to the problem.