Gonzo, like bullshit or beat or scumbag or love, might be a word best defined by half-synonyms and loose connotations. In this case: wildness, improvisation, manic flight, wheeling-and-dealing, bluffing, edgework, egotism.

Or a functional definition: handling something considered safe and sedate by consensus as if it was the last lap of the Indy 500 with a hand's breadth separating the top 2 cars.

Or a medical one: the adrenalin high produced by desperate staving-off of doom, all body systems functioning at 150% capacity like a cat that gains the strength of 10 men as it is being dragged to the bath.

A behavioral definition: trying to overcome dangers brought about by crazy actions with yet more crazy actions.

Though writer Hunter S. Thompson is the one who, by successful self-legendizing, has brought the word "gonzo" (of obscure origins) to fame, the mood of gonzo is present in many places. The movie Dr. Strangelove is an exploration of unsuccessful gonzo (if you consider nuclear eschaton an unsuccessful outcome). A college student deliberately holding off starting a paper until 2 hours before its deadline, then smoking weed while typing in order to get a combined panic/desperation/exhaustion/indifference/THC rush going is gonzo. The Marx Brothers might have something to say about it as well.

In a way, gonzo is a dark and sardonic version of the naive beat mania exemplified by Jack Kerouac's character Dean Moriarty. Hunter Thompson studied On the Road extensively, and who knows what lessons he took from it into the grim, ominous late 60s?