besiege

Besiege comes from Middle English besegen, bisegen, formed from the prefix be- ("around, about") plus siege. The word literally means to put a siege all around something.

The facts don't show a shift in meaning beyond its literal formation — from the start, it described the act of surrounding a place with a siege, a sense it still carries today, now often extended to being surrounded by anything overwhelming, like requests or worries.

Its clearest relative is siege itself, the word it was built directly on top of by adding the encircling prefix be-.