I heard my saxophone instructor say that there are 14 sizes of saxophones. I've only heard of 4 (soprano, alto, tenor, baritome). What are the other 10 types?
Today, saxophones are either in Eb or Bb, but C and F saxophones were produced in the early 20th century, so some of the standard horns had C and F counterparts. The most common of these was the C melody sax, which was a C version of the soprano that was popular because it used to read piano music (without transposition). There was also an F mezzo-soprano that has been used by a few performers (https://en.wikipedia.org/wiki/Mezzo-soprano_saxophone).
Some other obscure ones include sopranino (smaller than a soprano), contrabass sax (bigger than bass), and even the sub-contrabass sax (which was used by the Nuclear Whales Saxophone Orchestra).
There are other types that have been specially produced, like this G mezzo-soprano: http://thesax.info/blog/jessen-g-mezzo-soprano/
From Yamaha there were 14 types, but according to Yamaha now only 5 are in wide spread use: soprano, alto, tenor, baritone, and bass.
I tried to dig deeper to verify the claim of 14, but I could not find anything conclusive. Some sources claim the number comes from a patent that Sax obtained, but even that cannot be easily confirmed. In fact, I found this article about the saxophone patents: McBride, The Early Saxophone in Patents 1838-1850 Compared.
There is a image online which is supposed to be from Sax's original patent...
...no definitive claim could be made from just that picture, nor does it give names.
You can easily Google for a list of names of instruments in the saxophone family and find page like...
...but I'm not going to attempt making a list here, because I can't confirm anything other than the well known list of five above. At least with those I have seen them all with my own eyes!
There are 9 Saxophones in the primary family (marked with a dagger (†)). And there are as many as 15 types of saxophones, depending on how you define the word. The first four in the list are the most common. Instruments marked with an asterisk (*) are not always considered true saxophones and any one of them may not have been considered one of the 14 by the instructor.
The list is as follows:
Tenor† - In the key of B♭ (one octave below) - sounding range from A♭2 to E5
Alto† - In the key of E♭ - sounding range from D♭3 to A♭5
Soprano† - In the key of B♭ - sounding range from B♭3 to G♭6
Baritone† - In the key of E♭ (one octave below) - sounding range from C2 to A4
Sopranissimo†* - in the key of B♭ (one octave above) - sounding range from B♭4 to D♭7
Bass† - Key of B♭ (two octaves below) - Sounding range B♭1 to F4 (I think)
Sopranino† - In the key of E♭ - sounding range from D♭4 to A♭6
Contrabass† - In the key of E♭ (two octaves below) - sounding range D♭1 to A♭3 (I think)
Tubax* - Available in E♭, B♭, or C (all two octaves below) - sounding range of E♭ (the lowest) model is the same as Contrabass - D♭1 to A♭3 (I think)
C Melody - Key of C (one octave below) - sounding range B♭2 to F5
Connosax (aka Conn-O-Sax)*
Subcontrabass† - Key of B♭ (three octaves below!) - Sounding range B♭0 to F3 (I think)
Mezzo Soprano - Essentially a Soprano sax in F - Sounding range E♭3 to B♭5 - This may not have been counted separately from the soprano sax by the instructor