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Twelve fifty-nine (12:59) is the largest "prime time" of day on a 12-hour clock in hours and minutes. (If you include seconds, it's 12:59:59.)

When a child has lived for 1259 days, for the first time, their age in days, weeks, months and years are all prime. The child would be 179 weeks old, that is, 41 months, or 3 years. [Hartley]

The smaller prime of the smallest pair of consecutive primes, i.e., (1259, 1277), whose sum of digits is prime and equal. [Gupta]

Smallest emirp partial sum of lucky numbers. [Post]