  Previous studies of the nonlinear regime of the magnetorotational instability
in one particular type of shearing box model -- unstratified with no net
magnetic flux -- find that without explicit dissipation (viscosity and
resistivity) the saturation amplitude decreases with increasing numerical
resolution. We show that this result is strongly dependent on the vertical
aspect ratio of the computational domain $L_z/L_x$. When $L_z/L_x\lesssim 1$,
we recover previous results. However, when the vertical domain is extended
$L_z/L_x \gtrsim 2.5$, we find the saturation level of the stress is greatly
increased (giving a ratio of stress to pressure $\alpha \gtrsim 0.1$), and
moreover the results are independent of numerical resolution. Consistent with
previous results, we find that saturation of the MRI in this regime is
controlled by a cyclic dynamo which generates patches of strong toroidal field
that switches sign on scales of $L_x$ in the vertical direction. We speculate
that when $L_z/L_x\lesssim 1$, the dynamo is inhibited by the small size of the
vertical domain, leading to the puzzling dependence of saturation amplitude on
resolution. We show that previous toy models developed to explain the MRI
dynamo are consistent with our results, and that the cyclic pattern of toroidal
fields observed in stratified shearing box simulations (leading to the
so-called butterfly diagram) may also be related. In tall boxes the saturation
amplitude is insensitive to whether or not explicit dissipation is included in
the calculations, at least for large magnetic Reynolds and Prandtl number.
Finally, we show MRI turbulence in tall domains has a smaller critical
$\rm{Pm}_c$, and an extended lifetime compared to $L_z/L_x\lesssim 1$ boxes.
