Within the EB's, most companies will sponsor H-1B holders for green cards in the EB-2 or EB-3 categories in a 3-step process: PERM, I-140, I-485.
If an H-1B holder wants to get a green card based on work/accomplishments as a self-petitioner, the two main pathways are:
Eb-1A for extraordinary ability, helpful for people subject to long India/China backlogs
Eb-2 NIW, easier than O-1A, takes 2-3 years if you're not subject to India/China backlogs.
But unlike what the top level commenter wrote, the option for an H1B holder to become a permanent resident by their own action, rather then by their employer benevolence, is limited to very few (the green card lottery and the above mentioned narrow self-petition categories), and isn't a thing you can just do after a certain length of stay in the US.
I had got married by and had a green card through that before the backlog and what not would have got through on the h1b path, but if I hadn’t met my person I would have still ended up a permanent resident.
As far as I know, the only way to self petition for an extraordinary ability (EB1) based immigration, but that probably applies only to a small percentage of H1B workers. The rest are bound to their employer applying to grant the employees immigration benefits, which the employer may or may not do.