Tired as a mug

In need of sleep or rest.

The saying is derived from a coffee mug because it always has to have coffee in it to stay awake.

Man I've been working all day, I'm tired as a mug.

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Tired of your mug

Means that you are tired of adults nagging you to do something like washing the dishes or being done with society

A: I’m tired of your mug
Society: no you must do this