  In this paper we study a family of limsup sets that are defined using
iterated function systems. Our main result is an analogue of Khintchine's
theorem for these sets. We then apply this result to the topic of intrinsic
Diophantine Approximation on self-similar sets. In particular, we define a new
height function for an element of $\mathbb{Q}^d$ contained in a self-similar
set in terms of its eventually periodic representations. For limsup sets
defined with respect to this height function, we obtain a detailed description
of their metric properties. The results of this paper hold in arbitrary
dimensions and without any separation conditions on the underlying iterated
function system.
