I have never measured a cheap transformer that has a cheap soft iron core instead of carbon steel, maybe has a core that isn't even laminated with insulated laminations, and without enough turns on its windings to be efficient. Some guys have and insist that the peak voltage is the same as the RMS voltage due to the transformer compressing the voltage peaks.
I wouldn't say the cheap transformer saturates its core, I would say that its peak current operates it on the curve of its magnetic hysteresis graph, so it is approaching core saturation.