Anything tropical (Caribbean, Hawaii, Thailand, etc) will have plenty of creepy crawly critters. Even 5-star hotels will have the occasional roach or beetle. OTOH, I lived in Alaska for 3 years as a child, and mosquitoes were a serious problem in the summer (no pollen or other airborne allergens, however, and no snakes). Having worked a lot in the Phoenix area, I can tell you that seeking desert heat is no solution, either. I guess what I'm saying is that there may be no place you can go to escape insects.

It's worth noting that you can spend a lot of time around the Riverwalk in San Antonio without being bitten by mosquitoes. This is partially due to periodic spraying, but mostly due to a large bat population. The bats clear the air, but don't do much to rid the ground of roaches and other undesirables.