What bug bites could leave your skin itching and throbbing for days?

Any of them...: ...If there is sufficiently severe allergy to bug saliva. There are also, in a few places, bugs which actually deposit eggs under the skin, which would of course itch worse.

Bug Bites: Any bug bite can cause red, itchy and throbbing skin that can last for a few hours, or even a few days. If the redness is expanding, the site is getting warm or painful, then you might have a skin infection called cellulitis and a doctor needs to see it.