How big of a shed can I build in my back yard without a permit?

You need to comply with the Seattle rear yard requirements. Check with the city zoning administrator.
A phone call will give you the information.

Do not build without checking. It will be discovered and torn down.

Good luck to you.

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Each City is different. Many cities in California require permits if over 120 square feet. Check with the City. In fact, their municipal code may be available on-line.