I noticed that the generator of the second stable stem b is the square of the generator of the first stable stem a, in the sense that if take two copies of a and smash product them together you get b out. I'm wondering if there are any ( many) other exmples of this. What are the elements in the stable homotopy of spheres which a squares in the above sense?
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.
See the "ring structure" section for the stable homotopy groups of spheres: http://en.wikipedia.org/wiki/Homotopy_groups_of_spheres
This ring structure is well understood in the range that the homotopy groups of spheres are known. There are further operations called Toda brackets. The story goes on.