There are very many such languages. For example, Shona ha-ndi-za-ka-va-on-a means "I haven't ever seen them" (morpheme breaks included, in case you thought that might be a bare root). Greenlandic Nalunaarasuartaatilioqateeraliorfinnialikkersaatiginialikkersaatilillaranatagoorunarsuarooq "Once again they tried to build a giant radio station, but it was apparently only on the drawing board"; Imbabura Quechua micurcani "I ate"; Turkish yedim "I have eaten"; English "Leave!". There aren't very many one-word sentences in English. An interesting study would be to compare languages in terms of the number of 1-word sentences actually encountered in texts, since in these languages, even though you can construct 1-word sentences, typically there is more to a sentence than that.
Typically, the concept of agglutinativity is actually related to having the possibility of multiple affixes, which combine with significant freedom, independence and regularity. Because of the complexity of person-number-tense-aspect marking in Romance languages, they aren't considered to be "agglutinative" (nor is English), whereas Quechua and Bantu languages allow many affixes to combine, each with some semantic function, and you have few complex dependencies (such as "the 1s subject is -əg except in the perfective it is -ɬi). If you define "fully agglutinative" as a language with absolutely no (non-semantic) co-occurrence restrictions, shape irregularities, which also allows free combination of at least 3 affixes, there may be no fully agglutinative languages.