How many appeals can there be?
In the large states, there are three or even four levels of courts, while in some less populous states there are only two. There are important differences in the time limits, rules, and procedures depending on whether the case was brought in Federal court or state court. Different rules apply in each state. Also, often even within a state there are different rules depending on which court tried the case.
Most state courts require "finality" before an appeal -- in much the same way as Federal courts -- but there are many minor differences. A few states, such as New York, do not require "finality" for appeals. In such states, it is very common for parties to appeal decisions such as a denial of a motion for summary judgment (a motion that asks a court to decide the case on the law where there is no substantial dispute about the facts). In such jurisdictions, there are many more appeals.
We strongly recommend that, if wish to appeal your case, you should consult a lawyer experienced in appellate matters. Appeals are one area of the law that require the expertise of a lawyer who will help you assess your case. The average person cannot adequately handle them alone.