The queen lays around 1,500 eggs a day, peak season. Assuming 100% breeding success, in a typical month of thirty days she produces 45,000 new bees (1,500 x 30).
I'm guessing that the summer peak season lasts for four months, with an average of 1,500 eggs/day. For the remaining eight months, let's say there are four months of not-so-peak egg laying of 750 eggs/day on average, and four months of very little activity of 375 eggs/day on average.
If the figures are roughly correct, yearly egg production might be like this:
4 months x 30 days x 1,500 eggs/day = 180,000
4 months x 30 days x 750 eggs/day = 90,000
4 months x 30 days x 375 eggs/day = 45,000
An amazing 315,000 bees produced per year.
In winter, if the queen is producing low numbers of eggs or even none at all, the hive comes perilously close to extinction, because the bees will be dying more quickly than she is laying. During winter, the bees must live longer, or the hive would die off.
Looked at another way, if the queen lasts five years, she produces more than 1,500,000 eggs during her lifetime.