  We combine an axiomatics of R\'{e}nyi with the $q$--deformed version of
Khinchin axioms to obtain a measure of information (i.e., entropy) which
accounts both for systems with embedded self-similarity and non-extensivity. We
show that the entropy thus obtained is uniquely solved in terms of a
one-parameter family of information measures. The ensuing maximal-entropy
distribution is phrased in terms of a special function known as the Lambert
W--function. We analyze the corresponding "high" and "low-temperature"
asymptotics and reveal a non-trivial structure of the parameter space.
