We still talk quite frequently of someone being in their dotage, albeit often humorously, but the word nonage has fallen out of fashion. I came across it when I was reading Shakespeare's Richard III - when Edward dies and the citizens worry who will become king, one of them refers to the king's son, still a minor:
In him there is a hope of government:
That in his nonage council under him,
And in his full and ripened years himself,
No doubt shall then, and till then, govern well.
Nonage is made up of the negative word non (= lack of) + age and meant 'the state of being under full legal age'. Dotage is from dote + age, dote meaning folly or weakness of mind.
Now obsolete, the words boyage, child-age, adultage, infantage and barnage (from bairn, a Scottish or northern English word for 'child') are in the OED. As for what comes between nonage and dotage, well it's the also now-obsolete man-age, defined in the OED as "the age at which a boy becomes a man; the adult period of a man's life". (Needless to say, women don't get a look in in the dictionary.)