Can two neurons in the brain be connected more than once? Can two given neurons in the human brain can be directly connected more than once, either mutually or in the same or direction? Also, can the same neuron have transitive connections to itself (in order to amplify itself for example)?
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!