Are there any exactly solvable models in statistical mechanics that are known to have critical exponents different from those in mean-field theory, apart from the two-dimensional Ising model? I wonder about this because most easily solved models are either mean-field or do not exhibit a phase transition (Ising chain).
Yes, take e.g. the 6 and 8 vertex models. Mean field theory will typically fail in 2d models and so far we only have exactly solvable models for 2d systems. The book by Rodney Baxter that can be downloaded free of charge here, is compulsory literature for every student in this field.
If you want to expand your search to all critical exponents, I would recommend the following article:
Géza Ódor, Universality classes in nonequilibrium lattice systems, REVIEWS OF MODERN PHYSICS, VOLUME 76, JULY 2004
This contains lots of different systems that can be (are) modeled with lattices (e.g. Ising, percolation and interface growth). The contents list is actually pretty good starting point for simple listed classification. Some of them are exactly solvable, however most are not (try searching the word 'exactly' in the document).
There is more on this topic in this question: Where can I find a good classification for phase transitions?