Many studies have aimed to understand food webs by investigating components such as trophic links (one consumer taxon eats one resource taxon), tritrophic interactions (one consumer eats an intermediate taxon, which eats a resource), or longer chains of links. We show here that none of these components (links, tritrophic interactions, and longer chains), individually or as an ensemble, accounts fully for the properties of the next higher level of organization. As a cell is more than its molecules, as an organ is more than its cells, and as an organism is more than its organs, in a food web, new structure emerges at every organizational level up to and including the whole web. We demonstrate the emergence of properties at progressively higher levels of structure by using all of the directly observed, appropriately organized, publicly available food web datasets with relatively complete trophic link data and with average body mass and population density data for each taxon. There are only three such webs, those of Tuesday Lake, Michigan, in 1984 and 1986, and Ythan Estuary, Scotland. We make the data freely available online with this report. Differences in web patterns between Tuesday Lake and Ythan Estuary, and similarities of Tuesday Lake in 1984 and 1986 despite 50% turnover of species, suggest that the patterns we describe respond to major differences between ecosystem types.