Can a neuron make a synapse onto itself?
A synapse from a neuron unto itself is called an Autapse. Not a whole lot is known about them. Tamas et al. (1) give a summary:


  Van der Loos and Glaser (1972)proposed the word “autapse” to describe
  a transmitter release site made by the axon of a neuron and its own
  somatodendritic domain. In addition to their original Golgi study in
  rabbit neocortex predicting the existence of autapses, possible
  autaptic contacts have been described in dog (Shkol’nik-Yarros, 1971)
  and rat (Peters and Proskauer, 1980; Preston et al., 1980) cerebral
  cortex, monkey neostriatum (DiFiglia et al., 1976), and cat spinal
  cord (Scheibel and Scheibel, 1971). (...)
  Although autapses formed in cell cultures have been used extensively
  to study the physiology of synaptic mechanisms (Segal, 1991; Pan et
  al., 1993;Shi and Rayport, 1994), few proposals have been made for the
  functional significance of inhibitory autaptic innervation in
  vivo(neostriatum, Park et al., 1980; Aplysia buccal ganglia,White and
  Gardner, 1981).