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Yes you can. There are two paths, one requires more stairs than the other, but both include a fair amount of climbing, with a slight squeeze near the top. There is usually a decent line for accessing the dome, and it is up to you if you want to take your guide along- if you don't mind paying your guide for an extra hour, they are always happy to share their extensive knowledge with you!
Please visit The View from the Cupola article for more details about this experience.