It is well known that the maximum likelihood estimator of logistic regression does not admit a closed form solution, at least in the general case where the predictors are not binary or categorical. Whereas, ordinary least squares regression does have a closed form solution in terms of matrix inverses.
Is there a proof that logistic regression can't have a closed form solution or it is simply that none have been found after an extensive search? Proofs of the non-existence of things are presumably difficult in general. Is there at least an agreed upon definition of closed form in this context?