Are there any known models with limit cycles in their RG flow?
Such systems are quite possible, modelled copiously, the focus of a cottage industry, and have numerous applications. Beyond the Bulycheva & Gorsky review arXiv:1402.2431 that @Buzz links above, in his references you'd find particularly instructive papers. Foremost, in my mind, are LeClair et al.'s "Russian doll spin models":
A. LeClair, J. M. Román, and G. Sierra, Russian Doll Renormalization Group and Superconductivity, Phys. Rev. B69 20505 (2004) arXiv:cond-mat/0211338; Russian Doll Renormalization Group, Kosterlitz-Thouless Flows, and the Cyclic sine-Gordon model, Nucl. Phys. B675 584-606 (2003) arXiv:hep-th/0301042; Log-periodic behavior of finite size effects in field theories with RG limit cycles, Nucl. Phys. B700 407-435 (2004) arXiv:hep-th/0312141; A. LeClair and G. Sierra Renormalization group limit-cycles and field theories for elliptic S-matrices, J. Stat. Mech. 0408:P004 (2004) arXiv:hep-th/0403178.
S. D. Glazek and K. G. Wilson, Limit Cycles in Quantum Theories, Phys. Rev. Lett. 89, 230401 (2002); Erratum, 92, 139901 (2004).
E Braaten and H-W Hammer, Universality in Few-body Systems with Large Scattering Length, Phys. Rept. 428 (2006) 259-390 arXiv:cond-mat/0410417 [cond-mat.other].
T L Curtright, X Jin, C K Zachos, RG flows, cycles, and c-theorem folklore, Phys Rev Lett. 108.131601 , arXiv:1111.2649 [hep-th] and T L Curtright and C K Zachos, Renormalization Group Functional Equations, Phys. Rev. D83 (2011) 065019. arXiv:1010.5174 [hep-th], whose section IV gives you the minimal cartoon of it, below.
The renormalization of the dimensionless couplings $g$ and $h$ under a change in system (Russian doll Hamiltonian) size $L$ is given by $$ \frac{dg}{d\ln L}=g^{2}+h^{2}\ ,\ \ \ h=\text{constant} $$ with $h$ the time-reversal breaking parameter.
Assuming $h\neq0$, change
variables to $u=g/h$ and $t=h\ln L$. Then
$$
\beta (u)=\frac{du}{dt}=1+u^{2}
$$
and direct integration yields
$$
u\left( t\right) =\tan\left( t+\arctan u_{0}\right) .
$$
Thus the physics of the model repeats itself cyclically as the logarithm of the system size is changed, in evident evocation of nested Russian dolls.
There are numerous applications in spin physics, nuclear physics, and HEP ("Efimov states").
