Starting PhD without Bachelors If I'm already at a decent school but don't feel like graduating, and a professor at a university agrees to take me as a pure math PhD student, can I join the PhD program or must I get a Bachelor's. The universities I am interested in are medium-upper level math universities (Northwestern, Berkeley, UCLA, Ohio State, etc)
It is possible in theory and has happened in practice.  (As I have reported elsewhere on this site, my own former thesis advisor attended but did not graduate from Bronx Science and attended but did not graduate from MIT, and this did not stop him from being admitted to the PhD program at Princeton, where he received his PhD at the age of 22.)  

However in recent years university systems have gotten increasingly formalized. Especially at public universities, many bureaucratic requirements are taken seriously and are difficult or impossible to waive.  For instance at my state university (UGA), one needs a bachelor's degree to be eligible to enroll in our graduate program.  Speaking as the Graduate Coordinator of the math department at UGA, we would almost certainly not seriously consider an application from a student who could not meet this requirement.  On the one hand, it would be an uphill battle to get such a student enrolled.  On the other hand -- why won't they arrange to get an undergraduate degree?  Because they "don't feel like graduating"?  It is hard to understand why we should go out on a limb for an apparently feckless student when we have so many other applicants who really give a feck...so to speak.  Berkeley, UCLA and Ohio State are all state institutions, and I would expect (though I am not completely sure) their views to be similar to mine.

In my opinion the strategy of trying to bypass your undergraduate degree could only work if you are a student who is so exceptionally strong that it is clear to the very top schools that they want to have you.  Having said that...I got my PhD in math at Harvard, a perennially top-ranked program.  All of my fellow graduate students had (at least) undergraduate degrees.  On the other hand, more than one or two of them were under 21 when they arrived.  So in trying this maneuver you may be competing with absolutely brilliant 18 or 19 year-olds who were nevertheless feckfull enough to acquire undergraduate degrees.  I really don't like your chances.

Finally:


  and a professor at a university agrees to take me as a pure math PhD student, 


That's not how it works at all.  You have to get admitted by the department and the graduate school.  In pure math in the US, it is very rare for admissions decisions to be made by any one faculty member: they have to be made by the committee as a whole.