As the question is stated, there are actually no examples, at least assuming classical logic.

1. Classically, every small complete category is thin, i.e., a preorder with arbitrary meets. (https://ncatlab.org/nlab/show/complete+small+category#in_classical_logic)
2. By the adjoint functor theorem, any preorder with arbitrary meets also has arbitrary joins. (https://ncatlab.org/nlab/show/suplattice).

More concretely, if you have all meets, you can define an arbitrary join of a set $S$ to be the meet of all the upper bounds of the set $\bigvee S = \bigwedge \{y\in X | \forall s \in S. y \geq s \}$.

Of course if you change "all" to mean something more reasonable like "finite" or indexed by some other small cardinal, you can get many examples. 

Alternatively you can look at constructive mathematics, where you can add as an axiom that such a non-trivial small complete category exists. This is modeled by e.g., realizability toposes (https://ncatlab.org/nlab/show/realizability+topos).