Is it possible that two neurons stimulate each other in an everlasting two neuron circuit?
We refer to neurons that are connected to each other as having a "reciprocal" connection (however, the term is also used to refer to regional reciprocal connections in addition to neuron-neuron connections).
For typical neurons, much more than one input is necessary to reach action potential threshold, so you wouldn't expect reciprocally connected neurons to result in an "everlasting" response of the two cells. Pyramidal cells in neocortex, for example, are often connected to their neighbors, but that doesn't mean one always fires after its neighbor. Holmgren et al 2003 tested 542 pyramidal cell pairs from layer 2/3 of rat neocortex. 61 of those pairs were connected, and of the 61, 7 were connected reciprocally, and these were much more common among cells less than 50 microns apart.
There are, however, central pattern generators that can include reciprocal connections to create a repeating rhythm. These networks typically involve substantial inhibitory components, though, rather than just two interconnected cells.
Holmgren, C., Harkany, T., Svennenfors, B., & Zilberter, Y. (2003). Pyramidal cell communication within local networks in layer 2/3 of rat neocortex. The Journal of physiology, 551(1), 139-153.