Thanks to a link by @mins, The Guardian says (uncited):
The physicist made 25 further [totaling 27] balloon flights. In the mid-1930s, having climbed to an incredible 23,000m (14 miles) [...]
But, a NASA-published book says (also uncited):
Ultimately, Piccard made 27 balloon flights, setting a final record of 72,177 feet. They were altitudes that airplanes of the day could not touch.
72,177 feet is 21,999.6 m, which is 22 km and not 23, moreover the possible conversion from meter to feet in the NASA book makes it suspicious: the 22 km record by Anderson being misattributed to Auguste and converted to feet (more on that below), and the difference from The Guardian in the altitude makes both unreliable.
Jean – Auguste's twin brother – published a paper in 1938:
Auguste Piccard and his associates were able to make three flights with their original balloon (1931, 1932 and 1934). More flights were made with similar but larger stratosphere balloons from Russia and the United States. The most important of these flights was the National Geographic Society–U.S. Army flight of Stevens and Anderson (1935).
— Piccard, Jean. "Exploration by Balloon." The Scientific Monthly 47.3 (1938): 270-277. [emphasis mine]
Jean credits Auguste and associates with three flights, which agrees with Jamiec's answer. It takes one mistake in interpreting Jean's 1938 paper – see my bold emphasis – to propagate the incorrect claim of twenty-something flights by his brother Auguste.
This is also confirmed by comments from @mins (emphasis mine):
In this interview, in 1954 by the Swiss Radio, which is in French, Auguste Piccard talks about two ascents, and when asked if there were more, he mentions only from others, like "Americans":
- André Savoy:
You made no more attempts to the stratosphere after 1932, but there was the attempt by your brother Jean [...], any other?
- Auguste Piccard:
Yes Americans with Explorer climbed to 20,000 m, Russians who climbed a bit higher than me [...]