Apologies for the somewhat cryptic title.
For any first-order formula X, let ssm(X) be the size of the smallest finite model of X. By size I mean number of individuals. So, for example, ssm('Fa & ~Fb') = 2 because the smallest models of 'Fa & ~Fb' have two individuals. If X has no model, or only infinite models, f(X) is undefined.
Now, for any number n, let lssm(n) be the maximum of ssm(X) among all formulas X with length n. That is, lssm(n) is the largest size among the smallest finite models for formulas with length n. The specifics obviously depend on details of the relevant language: whether we have function symbols, identity, redundant Boolean operators, etc. So there are in fact several different functions here.
What is known about these functions? Are they computable? Can we say approximately what value they take for, say, n=30?