When an inductance is connected to an AC supply, the current in the inductor is

You don't specify which relay you're using so I'm taking this one as an example.

Rated voltage = 110V AC
Current = 21mA @ 60Hz
Coil resistance = 932\$\Omega\$

Now the total impedance

\$ Z = \dfrac{110V}{21mA} = 5240\Omega \$

That's the resistive part (932\$\Omega\$) with the reactive part at 90°. Then the reactance

\$ X_L = \sqrt{Z^2 - R^2} = \sqrt{5240^2 - 932^2} = 5154\Omega \$

Then, since

\$ X_L = 2 \pi f L \$

\$ L = \dfrac{X_L}{2 \pi f} = \dfrac{5154\Omega}{ 2 \pi \mbox{ } 60Hz} = 13.7H \$

That's a pretty high value, but if you do the same calculation for 50Hz, where the current is 24.2mA, you get a comparable value: 14.1H.

Does this type of geometry has any practical use?

Not that I'm aware of and, TBH I don't think it ever will. The significant mistake you have made is in assuming that a long solenoid wrapped around a "big" core will be significantly "different" to a single wire wrapped around the same "big" core. It won't because you have opposing currents in the smaller loops of the solenoid cancelling out as the solenoid snakes around the "big" core. Those cancellations do not contribute to a significant increase in flux in the "big" core.

The opposing currents won't do anything magnetically useful (that I can see) and the extra resistance (due to the smaller loop of the solenoid) just wastes power.

Sorry to be so negative but it looks like a waste of time BUT who knows, in the future something may be required that needs this. I can't say that - all I can do is analyse the efficasy of the idea in terms of magnetic and resistive properties.