Are the stable homotopy groups $\pi^s_i(\mathbb R P^{\infty})$ known for small $i$? In particular, I would be interested in the values for $i = 5,6$. A quick Internet search did not lead to anything.

2$\begingroup$ This might contain the answer: arxiv.org/abs/0710.5942 $\endgroup$ – Qiaochu Yuan Feb 10 '16 at 19:28

4$\begingroup$ In addition to the calculation that Matthias Wendy points out, Mahowald's memoir "The metastable homotopy of $S^n$" does a lot of calculation with the AtiyahHirzebruch spectral sequence for stable homotopy groups of projective spaces (because this calculation connects to the EHP spectral sequence). $\endgroup$ – Tyler Lawson Feb 10 '16 at 19:50
The following paper contains a list of stable homotopy of projective spaces in dimensions $\leq 8$:
 A. Liulevicius. A theorem in homological algebra and stable homotopy projective spaces. Transactions of the American Mathematical Society Vol. 109, No. 3 (Dec., 1963), pp. 540552
In particular, $\pi^s_5=0$ and $\pi^s_6=\mathbb{Z}/2$.
The middle column of table IV on page 82 of George W. Whitehead's "Recent Advances in homotopy theory" Regional Conference series in mathematics Number 5 lists the groups in dimensions up to 30 (including the 2 quoted by Matthias Wendt).