In bosonic string theory, excitations of open strings obey the mass relation
$$ M^2 = \frac{N-1}{\alpha'} \,$$
and this seems to imply that such a theory has an infinite tower of massive excitations with arbitrary integer spin. But what exactly is the spin of the $N$-th excitation, is it simply $N$ or something else?
I will give an example of the relation between $N$ and spin in a way that you can easily work out the general case.
For $N=2$ open bosonic string, in light-cone gauge, we have the following state:
$$ |\psi\rangle = (s_{\mu\nu}\alpha_{-1}^{\mu}\alpha_{-1}^{\nu}+v_{\mu}\alpha_{-2}^{\mu}) |0\rangle $$ note that are two non-trival representations of $SO(D-2)$, a symmetrical matrix and a vector. This two objects can be combined into one representation of $SO(D-1)$, the little group. This representation is a symmetrical traceless matrix, so is a spin 2 representation.
As you increase the $N$ you increase the number of operators $\alpha_{-1}^{\mu}$ as well as new operators like $\alpha_{-2}^{\mu}$, and so on. This will increase the possible spins we can make.
The spin operator in the transverse directions is given by: $$ S_{i,j}=-i\sum_{n=1}^{\infty}\frac{1}{n}(\alpha_{-n}^i\alpha_{n}^j - \alpha_{-n}^j\alpha_{n}^i) $$ so the maximum eigenvalue is obatined by $N$ applications of $\alpha_{-1}^i+i\alpha_{-1}^j$, given an eigenvalue $N$.
