ためしがない is used to express that something has never happened or is never the case. It conveys a strong sense that there is no precedent or example of something occurring. The word ためし(試し)means "precedent" or "instance," so the literal sense is "there is no instance of ~."
This expression is typically attached to the past tense of a verb (Verb[た] + ためしがない) and is used to emphasize that a particular outcome or event has not once occurred, often with a tone of frustration, resignation, or strong assertion.
Compared to simply saying ~たことがない ("have never done"), ためしがない carries a stronger emotional weight. It implies that the speaker has observed repeated failures or absences over time and is making a sweeping judgment based on that experience. It often sounds exasperated or emphatic.