my question is the same as in title. Is there an example of signal which after fft operation wouldn't have power on the it's y axis?
Is there an example of signal which after fft operation wouldn't have power on the it's y axis
Pretty much all of them. For the discrete case the Fourier Transform has the same units as the time domain quantity. For the time continuous case it adds $1/Hz$ since it's spectral density. So if your time domain signal is a voltage the units of the Fourier Transform would also be $V$ for the discrete case (which includes the FFT) and $V/Hz$ for the continuous case.
You can use the FFT to calculate the power spectral density but it typically takes a little massaging to have the units to come out to be Watts. In the case of a voltage signal you can square it but to be precise you would have to label it as "Power over a 1 Ohm real resistor" or something like this.
This holds for all field quantities: voltage, current, pressure, displacement, force, etc.