Appendix 3 of Ravenel's Green Book, http://www.math.rochester.edu/u/faculty/doug/mu.html#repub, has a chart of stable homotopy groups including much of the multiplicative structure. Figure A3.1 depicts some of this structure visually, while Table A3.3 lists the elements out by name and degree.
The next example of a square after \eta^2 is the element in the sixth stable stem, which is the square of the Hopf map \nu in the third stable stem.
Though stable homotopy is not multiplicatively finitely generated, you can consider Toda brackets, which are a form of higher multiplication in homotopy analogous to Massey products in cohomology, and it is known that the entire stable homotopy groups of spheres are generated by Toda brackets on the Hopf elements 2, \eta, \nu, and \sigma.
