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.
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.
You should answer the the following question: if you were in the Admission Committee of Berkeley, UCLA etc, why would you take the applicant mathwork21? How mathwork21 compares to hundreds, if not thousands, applicants who have perfect GPA, GRE score etc.
Nothing is impossible, there are always outliers. The question is if you are good enough to be an outlier. I can imagine a situation when this is possible. You simply need to win one or two IMO medals, and have a LoR from a Field medallist that says you are a genius.
You can join a PhD program without getting a Bachelors degree. Dr. Jane Goodall does not have a bachelors degree. She has one non-honorary doctorate.
However, if you are at a decent school and the professors like you and you like them, then getting a bachelors degree will be no sweat.
If the professors are all like mathworker21 is the future of math then they can credit by examination/life experience/whatever all requirements away and fast track you into their PhD program.
In fact, if they wish they can bypass the PhD, post-doc, and tenure process and make you a full professor if they can sense the future Fields prize winner in their midst.
Returning to reality. You are not a future Fields prize winner. They do not sense it. In order to have a traditional academic career, you are going to have to finish your bachelors degree.
There are a lot of things in academia that are possible to do in theory, in the sense that they have been successfully done by someone, but in practice are so difficult and risky to do, that the answer to a generic person asking about whether they can do them is, to an extremely good approximation, "no".
For example, it is in theory possible (or at least was, back in the 1940's) to get admitted to the math PhD program at Princeton on the strength of a 3-sentence long letter of recommendation, and to go on to write a PhD dissertation that ends up winning you a Nobel prize. But if an anonymous stack exchange user asks a practical question about whether they can hope to attain such things themselves, I would simply answer: "no, almost certainly you can't."
Now, what you are asking about is admittedly of a less tall order than winning a Nobel prize, but it belongs in the same category of achievements. For reasons explained in Pete's and others' answers, your plan will almost certainly not succeed (unless you know something about yourself that we don't, say that you have already been very successful in research, published multiple well-received math papers and managed to impress some well-known academics who are offering to help you get into a good PhD program without a bachelor's degree). Moreover, this plan is extremely risky, since once you find out that the plan didn't work your only option would be to go back to school and finish your degree, costing you time, money and emotional distress.
When weighing all of that against the option of just finishing your degree, however annoying and pointless that might seem to you, any sensible person would recognize that finishing the degree is the much more advisable option. Anyway, good luck!
Edit: there is another serious risk factor in your plan that occurs to me and that no one has mentioned yet. Even if you do get into a PhD program, it is far from guaranteed that you will end up graduating - even some very talented students sometimes end up dropping out (for miscellaneous reasons, for example loss of motivation). In the event that you don't finish your PhD, you will then be left without any degree, which could make it much harder to find a good job. So, I know your question was "can I" and not "should I", but it's something to think about...
I work in a research institution, there is a colleague who is trying to start a PhD without even finishing his graduate program. He is a smart man, our professors are willing to take him on board.
But it doesn't work like that. He can't be admitted without super strong evidence that he can indeed do research. Our professors don't have the power to admit him - they don't run the university. The university admission committee don't care how intelligence he is, they'd just look at formal documentation.
It's close to impossible be admitted into any reasonable university without any degree. You have to be an extraordinary exceptional, are you? If you're a rich man like Mark for Facebook, you can jump directly into a PhD degree!
How many academic papers you have with you as the first author? If you don't have any ... just finish your undergraduate degree...