I find more or less illusory to ask for non-algebraizable formal schemes which would not find into the scope of deformation theory. Indeed, a formal scheme $\hat X$ over $C[[t]]$, say, is noting but a family of schemes $(X_n)$, where $X_n$ is a scheme over $C[t]/(t^{n+1})$ together with isomorphisms of $X_n$ with $X_{n+1}\otimes C[t]/(t^{n+1})$.
On the other hand, I wonder whether classical examples of non-algebraic analytic spaces, or algebraic spaces, could be constructed in the category of formal schemes, but have no precise answer to give.
