What does limit transferable power through a transformer?
There are other things to consider. Here's the equivalent circuit of the transformer: -



The extra components that need to be considered are: -


Xp and Xs - these are leakage inductances and no-matter how hard you try to make a perfect transformer there will be some flux in the primary that just won't couple to the wires in the secondary - this is portrayed as an inductor in series with the primary winding and an inductor in series with the secondary winding. These limit the amount of current that can be taken from the secondary and can be compensated for by raising the primary voltage BUT that runs the risk of causing greater saturation losses.
Rc is the iron loss in the laminations - the core is conductive and without laminations, the core would act like a shorted turn so, laminations are used and this greatly reduces eddy currents circulating in the core but there is a power loss due to iron not being a perfect conductor and this will heat the core up and the greater you make the primary voltage (to accommodate Xp and Xs) the greater the heat generated.


I see both the above as limiting factors for power transfer.