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If you are a member of the Steam beta program, it might be worthwhile opting out of the beta program and reverting to a previous version of Steam (it should do this automatically) to see if this resolves your problem.
Another thing that is mentioned in the Steam forums, is allowing Steam to verify all of its files and then closing Steam properly (rather than letting Windows kill it as it shuts down).
If the window appears every single time you start up Steam, you're probably experiencing an issue with Steam realising it has updated.
It's worth noting that this is a well discussed issue with no official solutions available, there are plenty of different suggestions on how this can be fixed though and some of these work for some of the people experiencing the problem some of the time, so you'll want to try all of them if you really want to stop Steam from verifying every time you start it.
Here's a list and a guide with step by steps on how to possibly fix the stalling update/download for Dota 2 client. • Clearing out the Dota 2 files from your depotcache folder.