Formula 1 3 5 b7, so a C7 would be C E G Bb or 8753xx
Here are the CAGED open-position 7th chords:
C7: x32310
A7: x02020
G7: 320001
E7: 020100
D7: xx0212
And here are all the open-position 7th chords I can think of
A7: x02020 or x02223 or x02023
Bb7: x10131
B7: x21202
C7: x32310
C#7: x43101
D7: xx0212
Eb7: x65026 or x68089
E7: 020100 or 022130 or 020130
F7: 101211 or 101241 (fret 6th string with thumb)
F#7: 244320
G7: 320001 or 320031 or 323001
Thus there are 7th chords utilising open strings for every root except G# / Ab. Any others you can come up with?
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