What are some simple solutions to fill deep dental cavities?

Not simple: Unfortunately there is no "simple" solution for a deep dental cavity. There are better options for early cavities. The deeper the cavity goes the further it gets to the nerve. Once the bacteria has encroached upon the nerve, only two treatment options remain, which are not simple nor inexpensive. Your best bet for health and to keep cost down is to see your dentist on a regular basis.

See a dentist: The best and simplest solution would be for you to see a dentist and find out what your options are.

No simple solutions: Other than temporary sedative fillings which will last for only several weeks, there are no simple solutions. If fear is preventing you from seeking treatment, you can be be mildly sedated. If you are reluctant because of monetary reasons, then try to find a dental school in your area. Usually treatment can be provided by students or residents at a reduced cost. Treatment is supervised, of course.