Not even that much.
Because the AI sucks at fighting, in SP it's easy to continually increase the size of your army as long as there is free manpower. And while there are penalties for higher conscription laws, Service by Requirement is not that much of a burden on your industry.
But where the limitations come into play is with equipment. I've noticed that against a competent human player, assuming you don't lose the entire war in six months, equipment lost in combat and to attrition becomes the effective limitation on how much you can expand the army. Manpower losses might not be that severe, but even I might find it hard to expand Germany's army while also replacing 100k in lost infantry equipment, thousands of Panthers, hundreds of trucks, and who knows how much lost artillery.
This doesn't excuse how easy it is to get manpower into play, but I've felt constrained in my number of divisions (front line ones at least) at various times in some MP games.
Note that minors that use the generic tree tend to get tons of manpower they really shouldn't have due to Communist and Fascist NFs. This is where some of the ahistorical division spam comes from.
I think that the equipment thing is the major problem in the game right now. Not that it's bad, as such. I just don't think it's intended, at least not as such. It feels like the game intends for you to build units and then those units are, mostly, good to go until they die. If it's intended that you need to closely watch your production to ensure you keep your material stocks up then that feels like that should be an actual mechanic with a name and things to manage it not just background effect of the game running normally.