Sanitizable signatures (SaS) allow a (single) sanitizer, chosen by the signer, to modify and re-sign a message in a somewhat controlled way, that is, only editing parts (or blocks) of the message that are admissible for modification. This primitive is an efficient tool, with many formally defined security properties, such as unlinkability, transparency, immutability, invisibility, and unforgeability. An SaS scheme that satisfies these properties can be a great asset to the privacy of any field it will be applied to, e.g., anonymizing medical files. In this work, we look at the notion of γ-sanitizable signatures (γSaS): we take the sanitizable signatures one step further by allowing the signer to not only decide which blocks can be modified, but also how many of them at most can be modified within a single sanitization, setting a limit, denoted with γ. We adapt the security properties listed above to γSaS and propose our own scheme, ULISS (Unlinkable Limited Invisible Sanitizable Signature), then show that it verifies these properties. This extension of SaS can not only improve current use cases, but also introduce new ones, e.g., restricting the number of changes in a document within a certain timeframe.