Don't listen to physxl01. The circuit is an electronic ballast. There is no transformer stepping up anything. His comment is nonsense.
An electronic ballast doesn't even need a transformer at all. There is one small one in this particular circuit, but it handles no meaningful power, it's only a base drive feedback transformer to cheaply and simply drive transistor bases from the common mode output. It has three windings, one in series with the output tank driving the bulb, and two that drive each base of two transistors.
The large yellow thing is the output choke, and NOT a transformer. It is a couple to a few mH. It forms a resonant series LC tank that builds voltage until the lamp ignites. A the lamp shorts a parallel capacitor, lowering the resonant frequency and thus the voltage. This tends to be just a few volts above mains for a CFL.
The black bobbin inductor is probably, as you suspected, a 10mH choke. This serves as input filtering, and would be a common mode choke in a higher quality circuit.
The green toroid is found as base drive transformer. It's designed to limit current through the bulb by becoming saturated at that current which stops enough base current from getting to the transistors.
The green colored cores are ferrite with ridiculously high permeability. They are basically useless except as high frequency low current transformers, common mode chokes, that kind of thing. DC bias or anything that would store much energy in its magnetic field will saturate it (high permeability also means low energy storage) and make it worthless. In this circuit, it's being used as a saturable core by design.
The high permeability is how so few turns manage that. 20-50khz (what the ballast probably operates at) is fast, but not 3-4 turns fast, except on that high perm stuff.
Anyway, everything probably had a few mH of inductance. The green toroid probably develops 1mH or more from one turn (that's how high the permeability is).
The big ass output choke is much better as an inductor, and had comparatively higher saturation.