Note: this was originally a comment. I'm re-posting it as a full answer.
My sensei in college translated "Xは" as "as far as X is concerned", which often overlaps with "at least for X". But this "at least" is not referring to the quantity or amount as in "10 times or more", but it's more like refining the context as in "if we are talking about 10 times ..." or "as for 10 times ...".
一人じゃなかったら余裕で10回は行ってやるんだからな can be translated as literally as follows:
If I was not alone, for 10 times would I go there with ease.
Rephrasing it into more comprehensible sentence:
If I was not alone, I would easily go there for 10 times.
There is nothing to deduce about "less than 10 times" or "more than 10 times" because the topic of the sentence, marked by "は" is limiting the theoretical situation to "10 times".
But, if we look at the context, if the speaker could go there 10 times, could s/he go there 9 times? Probably yes, because in the context of going to the drink machine to refill drinks, the more often s/he goes the more embarrassing it would be, so if s/he thought s/he'd be fine going there 10 times, s/he would be fine going there 9 times or less too, wouldn't s/he? Now, would s/he be fine going there 11 times? We wouldn't be able to deduce that without further clarification from the speaker. S/he might already turn bright red after 10 times and who knows what would happen if s/he went the 11th times? So, in this context, it's actually "at most 10 times", and not "at least 10 times". Again, this deduction came from the context, and not from the use of "は" particle in the sentence.