How many college credits are meaningful to someone in the Army?

Aside from promotion points, the milestones of 60 and 120 (Assoc. and Bach degrees).
There's a difference between just getting a bunch of credits and getting a degree. The Army is good at handing you electives. Promotion points only will get you to SSG, after that it's time to concentrate in a discipline (preferably before).

The Army awards two promotion points per semester credit hour of civilian education up to 135 points for promotion to sergeant and 160 points for promotion to staff sergeant.

Depends on the person. For me, none of them were really meaningful while I was in. They were very meaningful once I got out and went after my actual degree.

(Don't be like me! Start going after your degree NOW so you can already have a Bachelor's when you get out and then either transfer all of that GI Bill to family or use it to go after a Master's degree. Plus... it REALLY helps when going for Senior NCO promotions.)