There are lots of good combinatorial examples that are not hard to define: e.g., posets, matroids, quasisymmetric functions, graphs. A few places to look: W. Schmitt, Incidence Hopf Algebras, J. Pure Appl. Algebra 96 (1994), no. 3, 299–330. W. Schmitt, Hopf algebra methods in graph theory, J. Pure Appl. Algebra 101 (1995), no. 1, 77–90. M. Aguiar, N. Bergeron, and F. Sottile, Combinatorial Hopf algebras and generalized Dehn-Sommerville relations, Compos. Math. 142 (2006), no. 1, 1–30.