When I say pursuing here, I'm talking about "levels" rather than actual chasing. I'm looking for shounen series in which you have either a boy who's struggling to reach or surpass a girl, or a girl who's trying to catch up to/surpass the main boy. Something like Danmachi perhaps for the former, not sure if I can think of an example for the latter (unfortunate since that's the one I'm more interested in). Even better if the two know each other. Like perhaps the hero has a childhood friend, and either she decides to follow the same path and catch up with him or he realizes at some point that she's beyond him and struggles to reach her once more.

Kekkaishi is essentially like that, with Yoshimori attempting to catch up to his childhood friend Yukine in the first half of the series. This relationship kind of flips halfway through when Yoshimori becomes stronger than Yukine.

Ragna Crimson has some of this in the beginning, though it's really not a focus.

Shounen Shoujo (AKATSUKI Akira) is also all about the main boy trying to surpass the main girl, but, uh, it's really different than what you want, I expect.

Completely forgot about Kekkaishi, which yes does indeed fit my request pretty well. Shounen Shoujo sounds interesting, though it is indeed a bit different from what I was looking for (much closer to Kekkaishi). I'll have to look at that Ragna Crimson. Are there any where there's a girl pursuing the boy? They can be co-mains, the main thing is that one aspect of the plot involves her deciding to try and reach and eventually surpass him, eventually evolving into rivalry. I've seen plenty of shounen and young-audience anime with a boy and girl main character, but I can't recall one with this sort of dynamic to the characters' relationship. Heck, now that I think about it, I can't even think of any series besides Kekkaishi where you have a boy and girl that are friends and active rivals. Usually they set up some different line of pursuit for the girl so they can clash at times but don't really "compete".

What made me think of this was an anime I recently started watching/rewatching (I'd seen the first half before). At the beginning the hero's childhood friend tries to convince him to pursue a path that she herself dreams of, but can't because she lacks certain requisite abilities that he possesses. After he starts down this path she becomes a rather minor character, but a few scenes later on suggest that his success inspires her to try and do something herself. At the same time tech is advancing toward allowing normal people to see and do what these people do. I so very much hoped that she'd decide to pursue the path herself, as it'd be cool on multiple levels. At the same time, a side character experiences the other version: his childhood friend supported him a lot early on but worried that he didn't have the skill/drive to make it; when they later meet she chews him out for not making any progress, which finally spurs him on to become serious, travelling on his own to find true strength; by the next time he shows up he's caught up to her, maybe more. I left really liking the dynamic between those two characters, but feeling like they really missed an opportunity on the other girl.

Hm, you're right that it's fairly rare for shounen to have a boy-girl rival instead of just a boy-boy rival...

Some others I've thought of now that I'm skimming the rivalry tag...

Ao no Orchestra has the girl trying to reach the level of the boy, but they're not actual rivals, more like friends. (The boy has another rival on the same level as him.)

Parts of the Pokemon Special manga has this, particularly the Ruby/Sapphire arc, though I'm not sure how much you'll get out of it if you're not into Pokemon...

I Want to Be a Receptionist of the Magic World! has a fierce rivalry between a boy and a girl, though it's a shoujo isekai romance, haha...