I just checked it for some place in the middle of South Dakota that is a flat plane with no tall structures (or any structures...) nearby [1].
It looks like right for today it is 12 or more about 40% of the time [2]. That includes satellites very close to the horizon which most devices won't use. Limiting it to satellites that are at least 10 degrees above the horizon, the max visible at once today there is 11, with 9 being the number visible that vast majority of the time.
Looking at other days it looks like the pattern of number visible at various times throughout the day repeats, but with a period that is slightly off from 24 hours, so the percent of time that a given number of satellites is visible is about the same every day, the the specific times that number is achieved slowly changes day to day.
