Sure... I'd say there's a number of reasons to go way past ~3K, or - conversely, not to even go that high.
Some steels are able to take and hold extremely high grit edges, without losing their bite, and seem to just keep begging for more. Akifusa's SRS-15 is a good example of such a steel, but there are others... In such a case, why not use finer stones that get the blade effectively sharper, and let you maintain it with less metal removal over the life of the knife?
As mentioned above, the added refinement has some benefits on some ingredients... I especially like knives I'm using around herbs to be rocking 6-8K+ edges, or higher if I'm using strops or naturals. I find it crushes them less than ~1K edges. Fruit & veg also doesn't oxidize as fast when cut with very high grit edges, which doesn't always matter, but sometimes it does. If you're pre-cutting ingredients for later use (Preparing veggies & dip for a party, for example.), this can genuinely help keep them fresher.
When learning sharpening, I also find it wise to have a finer stone in one's arsenal, as it makes the process of deburring easier, and also wastes less metal when you undoubtedly spend more time on the stone chasing an edge than an experienced sharpener does. This goes against the advice to newbies to 'Just pick up a 1k stone, until you know how to sharpen', but it's something I believe in strongly.
There's also an argument, that higher polishes on edges create less stress risers, which leads to longer lasting edges. I'd say that I can observe this on some steels... ZDP-189, and Shirogami / 10xx to name two, which is interesting, because they're at opposite ends of the spectrum in terms of alloy. With Shirogami/10xx in the kitchen, I'd possibly attest this to the added polish slowing acid erosion of the apex, more than stress risers. With 10xx in woodcarving tools, the highly polished edges definitely seem to stay sharp longer. On ZDP-189, coarse edges for EDC just don't hold as long as when I take it to 6K+. In the kitchen, same deal. Loses shaving sharp off of an 800 grit stone fast, but holds that shaving sharpness on 2-3 micron or finer edges really well... Steels with high vanadium & tungsten carbide counts seem to behave a bit differently, and hold coarser edges much better than pure carbon, or even some chromium carbide steels.
Conversely, some knives just can't make use of all that extra refinement... Victorinox, for example, uses a steel that's an utter waste putting anything higher than a ~1K edge on. It loses bite at higher grits, loses the ability to cut into a tomato quicker with a polished edge than a coarse one, and dulls so much during normal use that fine stones simply don't have enough grunt to efficiently bring the edge back from dull to sharp afterwards, so it's not like they can help extend the lifespan of the knife. Wusthof, Henckles, etc, are similar. Wustie's I'll give a strop with some buffing compound (A black tripoli, about 2500 grit. Same thing I'll use to help finish off Shun's.), but more because I find them to be an even bigger pain to deburr. I'll sometimes do the same on paring knives used for peeling from all of these brands.
As a rule of thumb, I would advise stopping at the grit level that can also touch-up the edge from normal dulling on that blade, and get it back to 100%. That's as high as that steel can 'practically' hold. If it requires more than one grit to re-do the apex once it's started to dull, you're possibly finishing it too high, or are pushing the blade too long before resharpening it.
For SRS-15, that touch-up point is something around a good Coticule, or a 1-Micron diamond strop on wood, or an 8-16K synth. For Aogami #1/#2 I find it's around the 3-4K point, or a BBW with medium slurry. For soft German/Swiss stainless, I find an 800 grit SiC stone is about perfect for touch-ups/use. For Ginsan or Shirogami I like pre-finisher J-Nats that are around the 8-10K level, a 6K synth, or a Hard Arkansas, depending on the sharpening style... These can all pull the blade back to 100% on one stone once it's trained, for my usage, once I notice the blade getting dull.
If you sharpen blades too finely, that final step won't necessarily be able to restore the edge on its own in 1-step. For example, 0.5 micron does not restore a dulled SRS-15 blade. You have to break down to at least 1-micron, to get the edge back from dull. Thus, if you finish at 0.5 micron, restoring that edge will be at least a 2-step process.
I just find that being able to touch-up in 1-step, on your finishing stone - whatever that grit is - tends to make the most sense. Whatever the finest grit you can touch-up on, is what your blade can actually handle in terms of refinement. What that grit is, varies depending upon the steel, HT, geometry, and how far past 'starting to feel dull' you push it.