Here is a quote from the list of episodes of one Japanese anime:
After Kazuto explains that he is unable to say what he thinks of Suguha until Asuna is rescued, Leafa agrees to help him in the Grand Quest this time around, convincing Recon to join them as well.
I am not sure I understand the meaning of this time around here.
Is it like "about this time"? If yes, then it doesn't make sense to me because in the story Leafa agreed to help Kazuto right this time, not "at some time around this time". That was about some dangerous adventure Leafa agreed to embark on with Kazuto, which was not going to be short or easy. That was not a mere appointment that Leafa agreed to show up for "around" some appointed time.
Could it mean "this time again"? That would make more sense to me as in the story Leafa had already helped Kazuto a few times before. However, I don't find the "again" meaning in the definitions of around given in Webster.
The meaning there that is closest to "again" is "again and again" like in "He always jokes around when he needs to be serious", in other words, it's about repetitive actions; but that's again not what is happening in the story: Leafa is not agreeing to be helping him "again and again". In fact, this is the last time she provides help for Kazumo in this story.
So, could it be some other meaning?