I've mostly worked as an IC and a manager in technical organizations outside of VC backed startup land. In those orgs I've found that most places consider 'Senior Engineer' to be a terminal level. By that I mean until you get to senior engineer it us up or out. In fact, as I manager, I've had explicit timelines on which I've had to get people to senior, or manage them out. Once you DO get to senior engineer they are happy to let you stay there forever because a) senior engineers are what keeps the tech org going day to day b) the staff+ path and management path are not just 'up' from senior - they are actually different jobs from senior requiring a different skillset. The downside of this is don't expect anything more than cost-of-living increases in pay.
I'm also a manager and came here to say the same thing. Senior is a terminal level at plenty of mature companies. If you're in the startup world, you'll probably face big pressure to take on a Staff+ or manager title, but you can certainly successfully push against that if you have good rapport with the leadership team. I knew several engineers with decades of experience who were very happy being senior engineers in that world. They had their niches and knew them well. One had been writing email systems since the 90s, for example.
