These limits usually take the form: "Because we observe [X], which occurs at rate [Y], the universe must be at least [Z] years old".
Many creationists respond that current rates ([Y]) might be different (slower) than past rates.
Indeed, it is possible that these rates changed — but under uniformitarianism, which is necessary for science to function, we must assume that rates did not change unless there is evidence for this change.
This article collects evidences that place a lower limit on the age of the Universe beyond the 6,000 to 10,000 years asserted by most Young Earth creationists (YECs) and the literalist Ussher chronology.
Moreover, these dating methods are not mutually exclusive: where their range, accuracy, and applicability overlap, the dates they produce agree with each other.
(For example, all dating methods for the age of the Earth agree on a 4.4-4.6 billion year-old world.) This is important especially because YECs regularly claim that radiometric dating is unreliable — yet radiometric dating is unnecessary to prove an old universe.