It is about a "conjecture" I heard (when I was student). There would exist an algebraic structure on the homotopy groups of spheres such that this algebraic structure would be the free algebraic structure generated by one point. In the wikipedia page "Field with one element", it is written that the algebraic $K$-theory of the field with one element is related (can be identified ?) with the stable homotopy groups of spheres. Is it the answer ?
I think the answer to the last question is no. As far as I understand, the "general linear groups over the field with one element" are supposed to be the symmetric groups. Therefore the statement that the algebraic K-theory of the field with one element can be identified with the stable homotopy groups spheres is essentially the Barratt-Priddy-Quillen theorem, which says that the group completion of $\coprod_n B\Sigma_n$ is the sphere spectrum.
As far as I can tell, the Barratt-Priddy-Quillen theorem does not yield that much explicit information about stable homotopy groups of spheres. On the positive side, I would like to mention that Smirnov apparently gave an explicit presentation of the $E_\infty$ page of the Adams spectral sequence as an $A_\infty$ algebra: "Description of stable homotopy groups of spheres in the language of A∞-algebras." (Russian) Uspekhi Mat. Nauk 51 (1996), no. 1(307), 171--172; translation in Russian Math. Surveys 51 (1996), no. 1, 171–172. But I don't know the details of this work.