A quantum state can't affect the wider world unless a measurement is performed, so in order for a quantum computation to be useful it must ultimately lead to or contribute to a measurement somewhere.
That being said, there are many quantum algorithms that transform quantum information, without measuring it, as part of some larger algorithm. Examples include subroutines like amplitude amplifications, the quantum fourier transform, reversible arithmetic operations, and magic state preparation.
An interesting corner case is catalysis, where a specific helper state is needed for an operation to work but the state is restored by the end of the operation. The state is not doing anything it just goes into and out of the operation unchanged, so how could the state be affecting any measurement? And yet if you change the state the operation will go wrong, which is ultimately revealed by measurements later returning different results.