You don't need a ticket to see a dinosaur in Berkeley. The University of California Museum of Paleontology says a limited number of fossil exhibits are on public display, including a Tyrannosaurus rex. They can be viewed free any time the Valley Life Sciences Building is open.
A research museum first
The UCMP calls itself primarily a research museum, and most of its collections aren't open to the public. Wikipedia says it's the main place where fossils collected around the state are stored. The original fossils came from the California Geological Survey of 1860 to 1867. Annie Montague Alexander was the museum's first benefactor and led some early expeditions.
The Valley Life Sciences Building was designed by George W. Kelham and completed in 1930, according to Wikipedia.
An early web pioneer
Wikipedia says the UCMP was one of the first museums with its own website, in the early 1990s. The site was nominated for a Webby Award five times. It also received a medal from the Smithsonian Institution. Today the museum runs education sites including Understanding Evolution and Understanding Science.
Visiting
Check the Valley Life Sciences Building's hours before you go. The museum is at 1101 Valley Life Sciences Building on the Berkeley campus.