log-likelihood vs. other monotonic transformations

Question

One of the most basic "tricks" in maximum likelihood estimation is to work with the log-likelihood instead of the likelihood itself, which works since ln is a monotonic transformation, and makes things convenient because many of the common distributions we work with involve exponentials and so taking ln simplifies things. Because of this, we get so used to automatically using the log likelihood. But I was wondering if there are any applications where people maximize different monotonic mappings of the likelihood function?