This phrase expresses strong feelings of frustration or annoyance. After months of dealing with delays, I've had a bellyful of the construction noise.
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Informal Usage
It's casual language, more likely to be used in conversation than formal writing. I've had a bellyful of his excuses; I just can't listen to any more.
Pro Tip 3/3
Metaphorical
The phrase uses 'bellyful' metaphorically to mean a large, often excessive, amount. She's had a bellyful of office politics and is considering quitting.