Why might you still want to write a resignation letter? Many experts believe that writing a formal or even a basic resignation letter when you're leaving your current job helps make you seem more courteous and professional. However, the real reason to write a resignation letter is to create a paper trail documenting that you have given notice (if your employer requires two weeks notice or some other amount of notice) on a certain date, and to formally kick of the process should there be any legal hiccups (quite rare) with incorrectly processing your final paycheck and last day of benefits.
Asking You to Stay Beyond Your Notice. In this situation, you won't necessarily get the promise of more money or a better title__but your manager will ask you, perhaps as a personal favor, to stay a few additional weeks or months until the company is able to hire and train a replacement.