To put it short: In which active research areas of (pure) mathematics no (or only minimal) knowledge in category theory is required ?

To put it long: I know almost nothing about category theory - but I know that I do not like solving problems and providing arguments by diagram chasing and very "high-level" arguments.

I'm currently getting near the end of my time as a student and want to start a PhD. And while I'm not lacking a skill in algebra (which has been taught to us in an entirely category-theory-free manner), and therefor assume I could manage with some amount of category theory, I'm just not a fan of "high-level" arguments. So it's more a thing of personal taste. Therefore I would prefer to specialize in an area which does not exhibit too much category-theory (ideally: none).
Now the thing is, when reading around in this forum it seems to me that also in areas, in which I wouldn't expect category theory to come up, like analysis, it actually does come up ("Lipschitz categories").

Thus I would like to know, which sub(sub)fields of big fields like PDE, or number theory can be dealt with only a very small amount of category theory. Especially: Are there any subfields of geometry that are category free ? I'm currently taking a course in differential geometry and to me it seems that category theory is (in disguise) almost everywhere, since our professor constantly explains how some diagram, that proves some assertion about manifolds is actually some category-theoretic notion.

When browsing for example through the work of Terence Tao suprisingly few diagrams and mentionings of "categories" come up, so it seems to me that certain areas of PDEs and number theory may fit the bill. But this perception may be due to that fact, that currently I understand almost nothing about what I read and therefore may have missed some categorical arguments.

Side question: Are there conversely any areas of applied mathematics that heavily use category theory ? Do there exist, for example, some applications of category theory on numerical mathematics ?

oneof the major points of category theory. Peter Freyd once quipped, "Perhaps the purpose of categorical algebra is to show that which is trivial is trivially trivial", which is how I interpret what you said. Anyway, I entirely agree with your sentiment. Further advice is: once you (OP) have found an area that is based on attraction, just learn what you need on a pragmatic, need-to-know basis. If you find that low doses of category theory is part of that, then you will learn a little of that too, but otherwise don't worry about it. – Todd Trimble♦ May 7 at 13:21