  The Efimov effect was first predicted for three particles interacting at an
$s$-wave resonance in three dimensions. Subsequent study showed that the same
effect can be realized by considering two-body and three-body interactions in
mixed dimensions. In this work, we consider the three-body problem of two
bosonic $A$ atoms interacting with another single $B$ atom in mixed dimensions:
The $A$ atoms are confined in a space of dimension $d_A$ and the $B$ atom in a
space of dimension $d_B$, and there is an interspecies $s$-wave interaction in
a $d_{\rm int}$-co-dimensional space accessible to both species. We find that
when the $s$-wave interaction is tuned on resonance, there emerge an infinite
series of universal three-body bound states for $\{d_A,d_B,d_{\rm
int}\}=\{2,2,0\}$ and $\{2,3,1\}$. Going beyond the Efimov paradigm, the
binding energies of these states follow the scaling
$\ln|E_n|\sim-s(n\pi-\theta)^2/4$ with the scaling factor $s$ being unity for
the former case and $\sqrt{m_B(2m_A+m_B)}/(m_A+m_B)$ for the latter. We discuss
how our mixed dimensional systems can be realized in current cold atom
experiment and how the effects of these universal three-body bound states can
be detected.
