100-sided dice roll problem When should I stop rolling if it costs $1 for each roll and I earn only the value of the final roll shown on a 100-sided dice roll? My intent is to maximise profit and I have unlimited rolls

The past is past and doesn't matter for your strategy, so after roll $i$ you have  the option of $\$X_i$ if $X_i$ is showing, or paying \$1 to get the random $\$X_{i+1}$, for a total of $\$X_{i+1}-1$.  The expected value of the next roll, and every future roll, is \$50-1=\$49.
Thus, if you are currently getting \$50 or higher you should stop, if you are currently getting less than \$49 you should keep going. If you are currently getting exactly \$49 you are indifferent in expectation and you need some other criterion -- perhaps you should toss a coin to decide.
