The legal limit for remaining tread on a vehicle tyre is 1mm. That means that if the tread is not at least 1mm higher than the grooves of the tyre, it is illegal to ride with that tyre on a public road.
Your personal opinion on this may differ, but we would suggest that you replace the tyre before the tread is that worn. Yes, tyres are expensive, but remember that a tyre is a safety-critical item. If the tyre does not grip as well as it should, hazards such as sand or water on the road surface become even more dangerous.
Also, in an emergency-braking situation, the tyres will not grip well enough, meaning your braking distance increases.
But tyre-tread wear is not the only indicator of when a tyre should be replaced. Even if you don’t ride often, the tyre’s compound also deteriorates with age. If the bike is parked in direct sunlight, the tyre will deteriorate even faster.
If you know the tyre is two or three years old, check the condition of the sidewalls as well as the tread. Check for small cracks where the sidewall meets the tread. Look out for cracks in the grooves between the tread patterns. Cracks there quite possibly mean that the general condition of the tyre has deteriorated to such an extent that you risk a burst tyre if you ride fast and the tyre gets very hot.
When in doubt about this, ask a tyre specialist for a professional opinion. Remember that the cost of a new tyre might be high, but it would still be lower than the damage to your bike if you cannot perform an emergency stop or the tyre bursts while you’re riding on the highway – not to mention the cost and trouble of personal injuries.