The largest insects that ever lived were dragonfly-like bugs of the order Protodonata, sometimes referred to as griffinflies (a model of which is pictured here). They had wingspans of nearly 2.5 feet across and huge mandibles, making them formidable predators.
Thankfully, they went extinct a long time ago at the end of the Paleozoic Era. Although insects can't grow quite that large in today's world, there are a few that are nevertheless large enough to give even seasoned entomologists a good fright.
This story was originally written in November 2010 but has been updated with additional information.