Titanium is expensive, although it is not rare and titanium dioxide is a ubiquitous component, it's part of such mundane articles as toothpaste, paper or plastics.
The reason for that is that the processes that extract titanium from titanium dioxide (which occurs naturally and just needs some inexpensive purification) are quite costly. The established industrial method for this is the Kroll process. It works by first chlorinating the titanium to $TiCl_4$ and then moving the chlorine over to some $Mg$, yielding $MgCl_2$ and pure $Ti$. Then follows a costly remelting process under vacuum to further purify the metal.
Recycling titanium scrap is even harder, and removing the other components from the alloys is not entirely possible, so recycling only yields titanium based alloys, not the pure stuff (which is not necessarily a problem since it's very rarely used in its pure form).
What specific chemical properties of titanium make it so hard to purify? For example, the chlorine process at the beginning of the Kroll process takes place with coke in a fluidized bed reactor, very similar to the reduction of $Fe$ in a blast furnace, but if the carbon takes care of the reduction, why is that not sufficient? Why is the chlorine route needed?
Research into titanium production dates back to the late 1880s, and the Kroll process, patented in 1940, made it feasible for mass production, but remains costly and has not been improved upon since, despite much effort. How would a chemist look for suitable reactants during such research?