How many double greens does the Old Course at St. Andrews have?

There are 7 double greens, leaving only the 1st, 9th, 17th, and 18th greens as single greens.

More about the double greens at the Old Course at St. Andrews

In the early days the holes were literally the same for the trip out and back. The players would play from different tees but over the same fairways to the same hole in the ground. Eventually, as popularity and numbers increased, the single holes become two holes and separated by a safe distance.
If you add the numerical values of each two holes that share a green it adds up to 18 – every time. 2 and 16, 3 and 15, 4 and 14, 5 and 13, 6, and 12, 7 and 11, 8 and 10.
The average size per putting surface is 22,267 square feet compared to 3,500 at Pebble Beach and 6,435 at Augusta.
The largest double green – 5 and 13 is an enormous 37,846 square feet. Imagine the house you could build on that footprint!
This article was written in 2010 prior to the British Open being played at St. Andrews. It gives an interesting insight in to the double greens
Below is a fun short video with Paul McGinley, 2014 European Ryder Cup captain, that gives us some idea of the size of the green. Especially watch from the one minute mark!

Size and Design

Large Greens: The Old Course features some of the largest greens in the world, with an average size of approximately 22,267 square feet. Some double greens exceed 37,000 square feet. This is significantly larger than greens at other famous courses, such as Pebble Beach, which average about 3,500 square feet.

Complexity

Varied Pin Positions: The greens are designed with folds, depressions, and pot bunkers, allowing for a wide range of pin placements. This variability can dramatically affect the difficulty of the course, with certain pin positions making it significantly harder to score well.

Strategic Play: The layout encourages strategic play, as players must consider not only their approach shots but also the potential for long putts and challenging slopes on the greens. This aspect of the course contributes to its reputation as one of the finest in the world.