Denver's climate records extend from 1872 to the present. This is a 142 year span.
Suppose that the probability of different weather outcomes doesn't change at all from year to year in Denver. Then, the chance that the record for any given day of the year was set in a particular year is 1/142. This means that the expected average number of record days per year in Denver is just a bit over 2.57 per year. The standard deviation (of this binomal distribution) is 1.60 per year.
Thus, in about 68% of years, 1-4 records are set for a given measurement, and 0-6 records are set in 30 out of 31 years (6 records is a 2.14 sigma event). A year with 8 or more new records would be a 3.4 sigma event that would happen only once every 1,667 years.
This simple analysis does not so easily apply to multiple measurements since the high and low, and precipitation in a given day, are not independent of each other.