You can't get GE (Guild Expedition) troops above your current age. However, you can get troops above your age from story quests pertaining to your progress on the continent map. In order to obtain the Grenadiers and Musketeers from the story quests, you'll have to age up to LMA, since there is one quest that requires you to build a Tower Ruin, which is an LMA decoration. You can't get colonial troops in HMA.
In response to the title, technically Bronze Age is when you can get units above your age, as completing the IA continent map rewards you with legionnaires, and getting into the bonus provinces should get you EMA heavy infantry, which is still entirely obtainable in BA, but the amount of camping required to achieve that would be inefficient.