They have forever to collect a balance from you. Furthermore they can add whatever penalties and fees they wish to increase that balance. Worst of all, they don't have to remind you or send you bills or any other notification. You owed it when you left the office. (There very well could be local laws that require notifications, but that isn't really the issue here.)
That dentist has every right to deny you service until you settle the account. Forever.
The question you want to know is how long can they sue you for the debt.
The statute of limitations on collecting that debt via court:
Which covers the rules on HOW LONG they have to collect the debt. Owing the money is one thing, but the rules and tools that you creditor has to collect the debt are another. You are probably worried about them suing you. But if you don't pay the debt (or settle in some way), that dentist can refuse to provide services to you, even if they write off the debt.
Ways you can be punished by your dentist for not paying the bill are:
Depending on your jurisdiction and/or type of debt, they typically only report it on your credit (if they are reporting at all) for 7 years. Even if you pay and settle the account, it will still be reported on your credit report for 7 years. The difference is how it is reported. They can report that "user133466 is a super reliable person who always pays debts on time". They can say "user133466 is a flake who pays, but takes a while to pay". Or they can say "user133466 is a bad person to provide services before collecting money, because user133466 don't pay bills".
Other people considering lending you money are going to read these opinions and decide accordingly if they want to deal with you or not.
And they can say that for 7 years. The idea of credit reporting is that you settle up as soon as possible and get your credit report to reflect the truth.
Getting a Judgement Against You to Collect:
One popular way to collect a debt to is to sue you for it. There, each state has a different time period on how long a creditor has to sue you for a debt.
If you pay part of the debt, that will often reset the clock on the statute of limitations, so be sure any partial or negotiated settlements state very clearly, in writing, that payment is considered payment in full on the debt. Then you keep that record forever.
They Can Hassle You For The Money:
There are other interesting points in the Fair Debt Collection Practices Act. See Debt collectors calling? Know your rights.
They can only contact you in certain ways, they must respond to you in certain ways, and they have limits on what they can say, who they can say it to, and when they can say it. There are protections from mean or vicious bill collectors, but that doesn't sound like who you are dealing with. I don't know that the FDCPA is a tool you need to use in this case.
Bottom line - You Should Pay:
You should negotiate your debt and try your best to settle up. From your post, both parties dropped the ball, and both parties should give a little. You should pay no or minor late fees, and the doctor should report your credit positively when you do so. If you both made honest mistakes, they both parties should acknowledge that and be fair, and not defensive.
This is not legal advice. But you owe the debt, so you should settle up. I don't think it is fair for you to not pay because they didn't mail you a paper. However I also do not think it is fair for the doctor to run up fees and not remind you of the bill.
Finally, you didn't bring up insurance or many other details. Those details can change the answer.