Apart from some Alpine instruments not listed here, the 3 row is the big daddy of the melodeon family. Normally just an extension of the 2 row melodeon with an extra key to play in and 12 basses. The most common by far in England is the A/D/G system, but elsewhere in the world G/C/F and F/Bb/Eb are also common.
Other systems with 3 rows include the C#/D/G which is a kind of hybrid between the diatonic melodeon and the irish chromatic system featuring fingering from both C#/D and D/G. Then there is the D/G/accidentals system which is basically an extension of the 2Â½ row system but as every keyboard I've ever tried with this system has been different from the previous, publishing them all would just be confusing. Finally, the B/C/C# or "British Chromatic System" invented by the late Jimmy Shand Snr. is a very versatile extension of the B/C system - usually found with Piano Accordion basses but sometimes with a 12 bass system, this layout has also been used by John Kirkpatrick and Tim Van Eyken.