Dock for 3 LG Ultrafine Thunderbolt monitors with M3 Max?

If I'm not mistaken, these would be 27" 5K monitors running in Retina "looks like 2560x1440" mode. While the Mac is sizing text and objects as though the monitors were 2560x1440 pixel ones, it is actually drawing text and objects in 5120x2880 pixel detail, and sending a 5120x2880 (5K) video signal to each monitor. An uncompressed 5K video signal requires four times as much bandwidth as an uncompressed 2560x1440 pixel one.

Your Mac has Thunderbolt 4 ports.

MacBook Pro (14-inch, M3 Pro or M3 Max, Nov 2023) - Tech Specs - Apple Support

MacBook Pro (16-inch, Nov 2023) - Tech Specs - Apple Support

As far as I know, Thunderbolt 4 does not have enough bandwidth to support three 5K monitors that are running at a 60 Hz refresh rate. Some Thunderbolt 4 docks might support connecting two 5K monitors – if you drop the refresh rate down to 30 Hz, but my understanding is that the usual limit is two 4K monitors, or one 5K/6K monitor.

Thunderbolt 5 potentially offers more bandwidth for connecting displays, but your Mac does not have Thunderbolt 5, and I'm not sure whether Apple has relaxed "the rules" for connecting displays even for those Macs that do.

P.S. – The reason that the LG monitors only have one Thunderbolt port (for the link to the computer) is likely that 5K video takes so much bandwidth. If there was a daisy-chaining port, it might only let you hook up drives and the like, not additional displays. And if those were Thunderbolt SSDs, you might get better performance if you hooked them up to a different host port that might have more free bandwidth.