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I know of three.

  1. Strength of current

  2. number of loops ( a cheeky way to 'increase' current?)

  3. radius

I'm having doubts about radius of circular loop affecting the magnitude of magnetic field ta centre of loop. As if magnetic field due to an infinitesimal current carrying element is inversely proportional to radius, and number of such elements would be proportional to the radius, the magnetic field at centre of loop would be independent of radius

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Thats because the magnetic field due to the infinitesimal current carrying element is inversely proportional to the $radius^2$ ( and not radius as you have mentioned). So when you multiply by the number of such elements ( which will give a radius factor) you still get a factor of radius (r) in the denominator.

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The magnitude of the field at the centre is- $$B=\dfrac{\mu_oNI}{2r}$$ andis thus inversely proportional to radius.

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