Well, when you first get to know a person, it seems likely that you will meet more, exchange messages more, than is normal for anybody simply because you are both unknown quantities and do your best to fill that hole. At least in my experience.
Maybe this is the time when one is more likely to step over some boundary, outstay a welcome, and so on. I tend to be upfront about it. I generally tell new people 'look, I have no idea how many replies to a text is too many, or when you will get bored of doing this or that activity, so you should just tell me when you want to stop.'
I find there is one of three possible outcomes.
1) The other is clearly disturbed by such frankness and no longer meets/calls/texts you. In which case, screw 'em.
2) They comply with your wishes.
3) They are far too polite to reciprocate your open-ness, in which case I look for a few cues.
When texting (I find this very easy), do their messages suddenly become shorter and/or noncommittal? "Yeah." "No." "Haha, okay." "Sounds good; see you then". I usually take these as a sign (based on experience) that the issue is over, or is settled, or at any rate, the conversation is over for now.
I find in person much more difficult than via texting on the phone, but I think people will usually yawn, or look at their watch, or say 'well....'. Again, in my experience, this usually seems to suggest a wrapping up of things. Alternatively, you can just decide how long something will last for:
How long shall we play video games for?
When is your bus home?
Etc