(N)DFA with same initial/accepting state(s) What is known about the class of languages recognized by finite automata having the same initial and accepting state?  This is a proper subset of the regular languages (since every such language contains the empty string), but how weak is it?  Is there a simple algebraic characterization?

Ditto for languages recognized by non-deterministic automata having the same set of initial and accepting states.
Finite automata in which the initial state is also the unique accepting state have the form $r^∗$, where $r$ is some regular expression. However, as J.-E. Pin points out below, the converse is not true: there are languages of the form $r^*$ which are not accepted by a DFA with a unique accepting state.

Intuitively, given a sequence of states $q_0, \ldots, q_n$ such that $q_0 = q_n$ either $n = 0$ or the underlying state diagram must have a cycle involving $q_0$. The latter case is captured algebraically by the Kleene star.