There is no numerical limit on the number of appeals a condemned person gets. First there is a direct attack on the judgment which goes to the CA Court of Appeals and automatically to the CA Supreme Court. Beyond that there are usually collateral attacks on the judgment that may take years. The most common of these is the writ of habeas corpus which may be based on new evidence (an appeal cannot be so based) and which goes first through the state courts and then to the federal courts. Again there is no limit of the number of writs that may be taken. In CA it is not uncommon for the process to take 15-20 years and even then execution is not certain. CA has about 600 people on death row but has only executed a handful in the last 10 years.
There's no limit, so the time and money spent on these is also unlimited. Since death is permanent and mistakes are irretrievable, this may be the right policy.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.
Of course there are limits to the number of appeals you get while on death row . . . but I agree an individual is not necessarily limited to a certain number of appeals and it can (and in my mind, should) take many years to resolve. Typically a death penalty case (and all other criminal cases) can be appealed in a variety of ways depending on the stage.
A motion for a new trial (before sentencing in CA); a direct appeal (based on the transcript); a state habeas petition (based on violation of state or federal constitution); a federal habeas petition (based on violation of federal constitution); and successor habeas petitions if permitted under AEDPA (1996 statute that curtails number of appeals for DP inmates).