I got a mini cactus a few months ago the one with the three green fins and
topped up with a red prickly ball shaped thing
Today i ripped the ball thingy off completely and i was wondering if another
one would eventually grow back to replace it??

No, the ball is a cactus without any chlorophyll. It was grafted onto the green
base to keep it alive. The understock may grow, that's all. If you want another
grafted cactus, you will have to buy one. If you did not let it dry out
completely between waterings, it was too much. Also, it may not have been
getting enough light. The grafted cacti are a novelty which is not too
practical.

Iris,
Central NY, Zone 5a, Sunset Zone 40
"If we see light at the end of the tunnel, It's the light of the oncoming
train."
Robert Lowell (1917-1977)