Finite free objects Free distributive lattice with any number of generators is finite. For example with 3 generator the lattice will have 20 elements.
Is there other examples of free objects that are finite and have at least ten elements?
 A: An example: any finitely generated free idempotent semigroup is finite.
A: One notion you might be interested in is that of locally finite varieties; they satisfy an apparently stronger finiteness requirement, namely: Every finitely generated algebra in it is finite. Indeed, this is equivalent to having all finitely generated free algebra finite.
A prominent example is the class of Boolean algebras.
There is a great deal of study of this varieties, in particular with the question of which of them are decidable, meaning that there is an effective procedure to determine if a first-order formula is a consequence of the axioms. The book The structure of decidable locally finite varieties by McKenzie and Valeriote has deep results in this direction. I suggest that you take a look into its zbMATH review.
A: One classical example, still unsolved in generality, is Burnside groups.
The free Burnside group $B(m,n)$ is defined to be the free group on $m$ generators with the relations $g^n=e$ for all $g\in B(m,n)$.
In general, it is unknown whether or not $B(m,n)$ is a finite group (though the answer is known in many cases).  In fact, we already do not know whether $B(2,5)$ is finite.
