The reason there are two Gs is that the first one is assimilated from a D. The etymological structure of the word is ex-ad-gero,¹ "from-toward-carry".
This happens for everything. Compare accelerate [ad-celero] / accommodate [ad-con-modus] / impossible [in-pot-s-ibilis] / intelligent [inter-leg-ents].
(Note that a doubled consonant in Latin is pronounced differently from a single one, as is still true today in Italian. English draws no distinction, which makes the etymological spellings likely to confuse English speakers.)
¹ This was already the form of the word in Latin. It looks malformed to me; I can't explain why exaggero is a first conjugation verb (with passive participle exaggerata) rather than simply being a prefixed form of its root (ag)gero, in which case the participle would have been exaggesta. Gero and aggero are both third conjugation verbs...
I think exaggero is a denominative, derived from the noun agger 'pile' (itself a nominalisation of aggero i.e. "what is brought"). As such it took the default (productive) first conjugation. You can see a similar effect in English denominative <to highlight> which takes the productive past tense suffix -ed in highlighted, not *highlit.
this word is usually spelled exaggerate (yes, with two g's.)