In 1962 Toda published his book "Composition methods in homotopy groups of spheres", which contains computations of $\pi\_{n+k}(S^n)$ for $k\le 19$ and $n\le 20$. The values of these groups are conveniently tabulated, and reproduced on the Wikipedia page [Homotopy groups of spheres](http://en.wikipedia.org/wiki/Homotopy_groups_of_spheres). The computations involve composition product, Toda brackets and the EHP sequence, as well as cohomology operations for the higher values of $k$.


I am fairly certain that more values of $\pi\_{n+k}(S^n)$ have been computed in the intervening years, perhaps with more modern methods such as the unstable Adams spectral sequence.



> 
> Does anybody know of an up-to-date table of known unstable homotopy groups of spheres, beyond the range shown in Toda's tables?
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Unstable homotopy groups of spheres beyond Toda's range
I don't know the answer to your question, but I asked Fred Cohen. He had this to say:


Most of the computations are in Mahowald's work
with the EHP sequence. This gives infinite
families at p = 2 with Rob Thompson's extensions to p > 2.


Specific extensions of 2-primary components in fixed stems
are in 


* N. Oda, On the 2-components of the unstable homotopy groups of spheres. II, Proc. Japan Acad. 53, Ser. A(1977), 215-218.
* N. Oda, Unstable homotopy groups of spheres, Bull. of the Inst. for Advanced Re-
search of Fukuoka Univ. 44 (1979), 49-152.
* N. Oda, Some relations in the 18-stem of the homotopy groups of spheres, Bull.
Central Res. Inst. Fukuoka Univ., 104, (1988), 75–83. ​


Brayton Gray gave families of elements of order $p^r$ in
$\pi\_\*(S^{2r+1})$ for $p$ an odd prime.


Example: It is not known whether there are elements of order 64 in $\pi\_\*(S^{11})$.


Similarly, it is not known whether there are elements of order 64 in the stable
homotopy of $\mathbb RP^{10}$, thus a potential counterexample to the Freyd conjecture.


Other than that, not much more in terms of specific computations
are in the literature as far as I know.


