My cat's appetite is absolutely insatiable. I give him a quarter cup of dry food in the morning and the evening, and often a third of a can of wet food as well. Am I starving this poor guy or is he just a cute little gluttonous furball?

Has your cat ben wormed? Cats/kittens can often have an insatiable appetite when they have worms.

He hasn't been to the vet in about 1 and a half years because he stays inside the house. Is it possible that he could catch worms inside the house?

My kitten is an indoor cat but i still make sure he is wormed and treated for fleas. Yes they can still catch both although the chances are much less of course by being indoors. You dont need to go to the vets to get wormers, you can also buy online. Personally i use DRONTAL for worming and FRONTLINE for fleas. You may not need to do it as often as you would an outdoor cat but it's best to do it all the same.
Here's a couple of links to online vet sites:

http://www.vetuk.co.uk/

https://www.petmeds.co.uk/default.aspx

I always buy them online at these sites as they are cheaper than alot/most vets.

Totally agree with you, 100% of the way,not kidding to!

I free feed my cats (I have 6!) leaving bowls of a premium dried food around the house and to be honest they never stuff themselves - just eat a little and often and none of mine are overweight - the eldest is just over 11 and the youngest almost 3.

I would as already mentioned worm your cat though - even though he is indoors.

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Kaz

For kittens, joy is for the asking and smiles are for the giving