As far as I know, the essential aspects of slavery is that the slave is the property of the master. If so, anything the slave acquires belongs to the master. So how could a slave ever purchase his/her freedom? Anything he/she acquired could be immediately taken by the master with no compensation. Was there some type of social taboo in Rome against taking such earnings, or was there actually a law that allowed slaves to hold property the master couldn't take?
Check this answer from /u/toldinstone
https://www.reddit.com/r/AskHistorians/comments/kgtxu5/in_ancient_rome_slaves_could_attain_great/