I don't have an absolute answer, and would appreciate some insight from someone who does know, but it's not quite that simple.
1) In a normal battle, I think the limit is three artillery units. I believe that you can then have additional non-artillery units committed, one of each type. I have seen more than three SUs in combat but don't ever remember seeing more than 3 artillery SUs.
2) In town and city combat, the number increases. I don't remember the town number, but in a full city hex, you can have up to 18 support units. Not sure of the mix of artillery and other units. And I have never come close to that many. However that is what the manual says and I know from my own experience that at least 9 SUs have been activated.
3) This doesn't count units directly attached to a corps unit. For example, if you have a heavy tank regiment attached to a Cav Combat Corps, the HQ supporting the Corps can then add its support unit on top of the corps unit.
4) I have seen mix reports regarding attaching Soviet SUs to Soviet Combat Corps (not HQs). I have been able to attach things ski battalions, Tank regiments, engineer units, and so on. However, I have tried repeately to attached artillery SUs to a Corps and not been able to - so either I'm doing something wrong (very possible), or it can't be done.
And my combats always (as far as I can remember) always involve more than one unit.
So the question would also be - is this per limit on each combat or each unit in the combat execution?
And how does this work if the defense - for example - comes from two separate armies? Is the limit a hard limit or does it double?