Are there any cats that don't shed?

I am referring to domestic cats and the answer is NO. They all shed fur as it is an entirely natural thing to do. Indoor cats shed all the year round while indoor/outdoor cats shed when there is more light as in spring and summer. Spring light stimulates a heavy spring moult.

Hairless cats shed much less - they can have whiskers that shed or break. If you hate hair around the home choose a hairless cat but there are downsides to these breeds which outstrip the downside of non-shedding in my view. Photo in public domain.

Associated:The wrinkly skin of hairless cats

The only other point worth making is that single coated cats shed less than triple and double coated cats as there's less fur to shed. The hairless cat breeds such as the Sphynx, Don Sphynx and the Elf (as three examples) don't shed except for their broken and short whiskers. Whiskers shed from time to time or they break and snap off for hairless cats. So there is some depositing of hair from hairless cats albeit much reduced compared to a class double coated standard domestic cat. But the demands of looking after a hairless cat are higher than for a normal cat. This cancels out the benefits of non-shedding in my opinion.

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Pointed and angular head of the Siamese. Photo: Frangipani Balinesecattery. Siamese have very close to the skin silky single coats and therefore there is less fur to moult during the lighter months.

Feline hair grows to a predetermined length as per the cat's genes. After growth there is a transitional phase called 'catagen' followed by 40-60 days of rest called 'telogen'. It then falls out (sheds) and gives way to new hair. As hairs are at different stages at any one time the shedding is gradual.