Can two neurons in the brain be connected more than once?
The commenter below alerted me to the existence of autapses, which apparently are chemical synapses between a neuron and itself. So those clearly exist. At the same time, it does seem that there is a mechanism that would suppress self-synapses: a protein called Dscam that acts during neuronal development. Dscam is a membrane protein that can form a high number of isoforms through alternative splicing and thereby provides an identity to the developing neuron to prevent forming connections to itself. 

I suspect that what is more common is for a neuron to be connected to itself indirectly. That is, neuron 1 stimulates neuron 2, which stimulates neuron 3, which stimulates neuron 1.

EDIT: Just heard about this article, which shows that two neurons can be connected via two different synapses!