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Following the recent advances in the study of groups of circle diffeomorphisms, we describe an efficient way of classifying the topological dynamics of locally discrete, finitely generated, virtually free subgroups of the group Diff+ω(S¹) of orientation-preserving real-analytic circle diffeomorphisms, which include all subgroups of Diff+ω(S¹) acting with an invariant Cantor set. An important tool that we develop, of independent interest, is the extension of classical ping-pong lemma to actions of fundamental groups of graphs of groups. Our main motivation is an old conjecture by Dippolito [Ann. of Math. (2) 107 (1978), 403-453] from foliation theory, which we solve in this restricted but significant setting: this and other consequences of the classification will be treated in more detail in a companion work (by a slightly different list of authors). [ABSTRACT FROM AUTHOR]