I was just reading this article by Nikolaus, and at the beginning of Section 4, was surprised to read that there is no explicitly known set of generating acyclic cofibrations for the Joyal model structure.
Generating cofibrations, as pointed out in the article, are given by the boundary inclusions $\partial \Delta^n \to \Delta^n$, of which there are countably many. The problem is apparently with finding generating acyclic cofibrations.
Now, a fibration in the Joyal model structure is an inner fibrant isofibration. To get inner fibrancy, we just need to include among our generating acyclic cofibrations all inner horn inclusions $\Lambda^n_k \to \Delta^n$, of which there are countably many. To get isofibrancy, I had assumed we just need to include as a generating acyclic cofibration a map $\cdot \to \mathbb{I}$ from the terminal category to the nerve of the "walking isomorphism" -- the codiscrete category on 2 objects.
But for some reason, it's apparently not that easy. Why not?
The trivial cofibrations are the maps that lift on the left against all fibrations, that is, against all maps which lift on the right against the inclusions of inner horns and of a point into the walking isomorphism, that is true. But that's not what it means to say that the latter set generates the trivial cofibrations-for this, every trivial cofibration would have to be a transfinite composition of pushouts of coproducts of the maps you've listed. The latter are anodyne extensions, which do form the left half of a weak factorization system by the small object argument, but which need not coincide with the trivial cofibrations. This is a general problem with Cisinski model structures-you can see more on the nLab page of that title.