Garlic, salt, olive oil. Real aïoli is not mayonnaise with garlic in it — it is its own beast entirely.
The Provençal peasants' sauce, older than French cuisine itself. Traditional aïoli is JUST garlic pounded to a paste in a mortar with salt, then olive oil worked in drop by drop. The eggy shortcut version we all make at home is called 'aïoli bâtard' — the bastard aïoli. Accurate.
Pound 4 fat garlic cloves with 1 tsp coarse salt in a mortar until a completely smooth paste — this is not optional.
For traditional: work 200ml good olive oil in drop by drop, grinding with the pestle. Ten minutes of patience for a thick, garlic-hot paste.
For the bastard version: whisk 2 egg yolks into the garlic paste first, then stream in 200ml olive oil like mayonnaise.
Finish with a squeeze of lemon and rest 30 minutes for the garlic heat to mellow.