Absolutely. I can't speak for other cultures, but the Hindu world had a system of cosmogonic cycles consisting of yugas (eras), mahāyugas (great eras), kalpas (eons), and mahākalpas (great eons), beyond more standard, smaller measures of time.

A mahāyuga consists of a cycle of four yugas, with each one being shorter in duration than the previous in a 4:3:2:1 ratio. The first, the satya yuga, is equivalent to 1,728,000 years, while the last, the Kali yuga, spans for 432,000 years. The length of the mahāyuga is then 4.32 million years. This represents the complete lifespan of a deva (divine being).

So 1000 of those, otherwise known as a kalpa, is a day in the life of Brahmā, the creator god. That's 4.32 billion years, then his night is another 4.32 billion years, for a combined 8.64 billion year cycle of creation and destruction of the universe. Brahmā goes on to live for 100 of his years, composed of 360 of those days + nights, for a grand total of somewhere in the ballpark of 313.5 trillion human years. This is said to be one breath of Mahāviṣṇu... and so on.

There are different interpretations of the numbers involved here, but even the most conservative versions are massive. The point is not so much the specifics of the numbers, but the vastness of their expanse and the fact that they were included in the Viṣṇu Purāṇa, which was probably conceived before 1000AD, and possibly as far back as 700BC, but certainly before the modern era of big data.

While these numbers certainly weren't derived from empirical observations, they still had a functional role as big numbers. Their significance is that they compel the contemplative to urgency in seeking liberation from the neverending round of rebirth. When faced with the unfathomable infinitude of existence and suffering, one can't help but feel a disconnect from the frivolities of mundane life. I personally find it to be a beautiful conception of the cosmogonic cycle.

So there's one instance of huge numbers in antiquity. I'm curious to read of other examples.