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)?
I guess the answer depends on what you mean with "connected more than once".
Initially, being connected is a strictly binary relation: either two neurons are connected or not. (The question may be: directly or not. Indirectly, each pair of neurons is connected, as I assume.)
If you consider "being connected more or less strongly" - which is not a binary relation anymore - you may consider the degree of connectedness as a natural number 1, 2, 3, ... and in this case, you might want to count synapses between two neurons. And surely, there can be more than one:
»An important experimentally observed feature [...] is the distribution of the number of synapses from one neuron to another, which has been measured in several cortical layers. All of these distributions are bimodal with one peak at zero and a second one at a small number (3–8) of synapses.«
If you want to consider pathways between two neurons A and B - shortest sequences of connected neurons starting with A and ending with B - then the answer is "yes, of course" again:
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!