I am working a project now with an env texture that is 16385 X 8110 using Cycles nodes. It runs fine but my GPU is an Nvidia GTX 980. I may tray to cut it down a bit but this project is designed to take advantage of high powered PCs.
I am pretty sure I noticed the Everest project loading in tiles dynamically. You could end up with much higher resolution that way but you also have to deal with looking at blurry tiles for a second while it loads in the high res tile.
So your answer depends entirely on your target audience. Gaming rigs can handle some huge textures but if you want it to work on mobile device, it needs to be smaller.
You also have the option of Resource Conversion in your Project Manager. It will automatically make smaller texture sizes that will load on less capable devices. There are some URL parameters that will allow web player project to load compressed textures. Attributes.
Here, you can see the results of my conversion. The min50 is the scaled down version.