The way that many algorithms would deal with such a desire is to incorporate the measurement at a more fundamental level, essentially making it part of the 'while' condition. i.e. you have an output qubit that is 0/1 for computation complete or not, you measure it, and decide whether to continue or not. Because that's a bit of classical processing, it doesn't have to be reversible, and you avoid the need for infinite space.
For example, many quantum algorithms only have a finite probability of success. For example, search or Factoring. In both of these cases, you know if you succeeded, and so there's an additional bit of classical logical that says "repeat until successful" i.e. a while loop.