Should I ask my dentist to pay for my tooth replacement because of a cronic infection which wasn 't fixed the 1st, 2nd, or the 5th visit over the course of 8 years?

It depends on whether  the dentist committed malpractice. Malpractice is literally "bad practice"; a medical professional can be liable for injuries (including pain and suffering) and medical costs when he or she provides medical care which does not live up to currently accepted standards of care, such as through negligence (unreasonable carelessness). Basically--and to oversimplify--if the average reasonable dentist would have detected and corrected the problem during your visits, then the failure to do so may be malpractice.

That's the law. As a practical matter, you also need to factor in that malpractice suits--you'd have to sue if the dentist will not voluntarily pay for this--are among the most expensive types of lawsuits to bring. For example, you'd need a dental expert witness (who generally are expensive) to testify on your behalf to establish that this was malpractice. Depending on how much money is at stake, it may not be wortwhile to seek legal redress.