The USS Enterprise NCC-1701 appears to have a clear dome over the bridge:
There is a scene in the pilot that zooms in and peers into the bridge from space:
Does the crew have the ability to see through the dome into space?
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No. What you see is a retro filming style. The Dome was opaque.
I've discovered a scene in S03E02 to prove it:
Thanks to @TangoOversway for the screenshot idea.
We never see the complete bridge ceiling from inside in any episode of TOS. We see it for the first time in TAS and it is definitely opaque.
It could have been that the producers realized that showing space on the inside ceiling was beyond the FX technology of the time and decided to give up on it after the pilot.
I believe that the pilot episode zoom in may have represented a more symbolic approach showing the audience that the bridge was on top (ie most important part) of the ship.
(yes, this is rather late)
There would, in reality, be zero need to have a transparent ceiling on a starship bridge. The bridge will very likely not be anywhere near the top of the ship in any case - why put your command center and senior personnel in such an obvious, exposed location? One good phaser hit and the ship is decapitated.
Oceangoing vessels, aircraft etc. put the bridge on top so they can see where they are going - ships that rely on sensors (like submarines) don't. A submarine's bridge is in the main pressure hull below the sail, and it's only there because the command officers need quick access to the periscope and occasionally the top of the sail. The latest generation of subs (Virginia class) don't use optical periscopes and the bridge is now more-or-less in the center of the boat on deck 2. Of course, this wasn't the case in 1966 (or even 1987).