There are better grouts, but eventually, they will all need cleaning. How about Luxury Vinyl Tiles? The click lock type are pretty waterproof. And some are glued down and can be grouted to look like real tile, but they don't have to be. I don't think the grout they use is the same as tile grout, but it may need cleaning eventually too. There are some very realistic looking products now and will feel a bit warmer on the feet also.

Epoxy grout. It gets dirty, but the dirt stays on the surface instead of getting ground into the grout.

I hate cleaning grout too. In our latest reno, our contractor used a new kind of grout (epoxy?) that is less prone to absorbing dirt. But it is more temperamental to work with, so be sure to read all of the instructions. We also chose a grout in the color "natural", which closely matched the gray tile we were installing. (Darker grout generally won't show dirt the way lighter grout does.) So far, we have seen no evidence of soiling or traffic patterns.

I've used epoxy grout in our last several jobs and it is much harder to grout with. It needs to be immediately wiped from tile or it is very hard to remove. It seemed on the porcelain we used it was even worse than the ceramic tile we used it on. And the finish is grittier so I find anything on a horizontal surface holds dirt worse, even if it doesn't actually soak in. So I wasn't as pleased as I had hoped to be.

They make tiles that are 5 feet by 12 feet. More commonly you can find 48"x48". That will eliminate a lot of the grout lines.

We just had our master bath floor redone in 2'X2' tiles, with a 1/16" grout line, so there is nothing much to get dirty. The tile guy then did epoxy grout, and I can confirm it is a tough job: he spent 10 hours grouting about 400 sqft of floor and walls. But when it is done by a pro, there should be no residue left on the surface.