Using the metric of the first quote below, it would appear that an adult dragonfly could conceivably eat anywhere from 3000 to 18,000 insects in a summer - but remember that most adult dragonflies don't last the whole summer and the amount would vary by individual and species. These following quotes also point out that dragonfly larvae also eat a lot too.
'Dragonflies, which eat insects as adults, are a great control on the mosquito population. A single dragonfly can eat 30 to hundreds of mosquitoes per day.'
'In their larval stage, which can last up to two years, dragonflies are aquatic and eat just about anything-tadpoles, mosquitoes, fish, other insect larvae and even each other.'
'There are more than 5,000 known species of dragonflies, all of which (along with damselflies) belong to the order Odonata, which means "toothed one" in Greek and refers to the dragonfly's serrated teeth.'
'Adult flying dragonflies eat other smaller flying insects, esp. mosquitoes and gnats.'
'As flying adults, dragonflies live only a few weeks, but if you include their underwater stage as nymph, their lives span from 1 summer up to 3 years'
'Dragonflies have existed for over 300 million years! In fact, they were here before the dinosaurs, and are among the most ancient creatures still populating our planet Earth.'
'...the air-breathing, winged adult never lives more than a
few months - 2 or 3 months is probably the average, with a few surviving a bit longer.'
'Adult dragonflies eat other flying insects, particularly midges and mosquitoes. They also will eat butterflies, moths and smaller dragonflies. There is one Asian species which eats spiders from their webs! The larvae, which live in water, eat almost anything living that is smaller than themselves. The larger dragonfly larvae are known to catch and eat small fish or fry. Usually they eat bloodworms or other aquatic insect larvae.'
'At the shortest, a dragonfly's life-cycle from egg to death of adult is about 6 months. Some of the larger dragonflies take 6 or 7 years! Most of this time is spent in the larval form, beneath the water surface, catching other invertebrates. The small damselflies live for a couple of weeks as free-flying adults. The larger dragonflies can live for 4 months in their flying stage. In Britain, lucky Damsels seldom go more than two weeks and Dragons more than two months. Most Damsels rarely go more than a week, and Dragons two or three weeks. They die from accidents and predation, and large numbers from starvation - in poor weather neither they nor their prey can fly'
'The biggest wingspan of a living dragonfly is the Central American Megaloprepus coerulatus with a wingspan about 19 cm. This is a thin, long-abdomened damselfly. The bulkiest dragonfly may be Petalura ingentissima from Australia (female wingspan to about 16 cm), a central African Anax species or a reported, but apparently uncollected, aeshnid from Borneo. Perhaps the smallest Dragonfly is Nannophya pygmaea from east Asia including Malaysia and Japan. This species is only 15 mm long with a wing span of about 20 mm.
In prehistoric times dragonflies were much larger, the largest flying insects ever. The largest member of the extinct Protodonata was the Permian Meganeuropsis permiana with a reconstructed wingspan (based on fragments, scaled to complete fossils of similar animals) of about 70-75 cm.'