What are some interesting examples of accessible $\infty$-categories which are not presentable and not small?

By interesting I mean a category which comes up naturally in a certain context and in a way which doesn't allow working around it.

I prefer examples from derived algebraic geometry as these will most likely be more understandable to me but any interesting example will do.

Edit: As Dimitri pointed out in a comment there are already examples in ordinary categories (e.g. the category of fields). To focus my question I'd like to add the restriction that the category should not be $1$-truncated.