If I'm playing a G9, then want to add a major 7th to it, the notes would be: G B D F A F#
Could we notate that as G9 add b15?
Is there a more correct name for this chord?
Yes, you could call it G9♭15, but it would be a pointlessly irregular usage of the chord naming conventions that music theorists (mostly) agree upon. So in reality, no one uses that name, not least in part because diminished fifteenths absolutely suck.
The notes you name don't really have a standard, agreed-upon chord label, because that chord contains the m7 and M7 over the root. This sound isn't used very often. I think there are a couple of unambiguous ways to name the chord that are easier to understand. G9(addmaj7) is one option, depicting the chord as a dominant chord with an extra-spicy major seventh thrown in for kicks, and another way is Gmaj9(add♭7), realising the chord as a major chord with an added minor seventh.
Depending (quite heavily, actually) on how the chord seems to be functioning within the piece as a whole, one of these two general systems may be better than the other. If the chord is in the key of C major, it might make a lot of sense to analyze it as a G dominant chord, as opposed to as a G major chord.
I chose the nomenclature that I did for some specific reasons. Firstly, both styles construct a 9th chord of some quality, viewing the other note as a tension, albeit a strange one. Technically, the parentheses should render the "add" part of the chord label unnecessary, but I left it in to be clear that the entire G9 chord is present in the voicing.
For those wondering, Gmaj9♭7 is likely to be read as a mistake, since accepted standard practice dictates that an altered note in a chord symbol modifies the existing note within the basic chord rather than adding an entirely new note to the chord (incorrectly implying [G B D F♮ A]). This is another reason why I used "add" and a set of parentheses to explicitly demand an added note.
In Gmaj9(add♭7), I also chose to label the minor seventh as "♭7", since most jazz tensions use accidentals to raise or lower the note. "♯7" for the major seventh is ugly, because most of the time in music, ♯7 is enharmonic to 8, introducing unnecessary confusion. Most musicians would recognise "♭7" as the minor seventh of a chord, but few would instantly recognise "♯7" as major seventh.
An argument could actually be made for Gmaj9(add♮7), but I chose to use "maj7" because most musicians are already used to "maj7" in chords like Gm(maj7) which refer to F♯ as "maj7" rather than "♮7".
Why, though, would you need a chord label for this chord? This question is bordering on the kind of chord where the best answer is to simply write out the desired voicing on sheet music. Seeing any of the options I wrote on a lead sheet, most musicians would be confused, at least temporarily. It's not that there are no situations in which this chord would be idiomatic; it's that the situations which use this kind of chord realise the futility of chord labels. There's almost no overlap between the music that uses lead sheets with chord symbols and the music that uses that chord.
If you do want to talk about using the 15th, the only person I've ever heard talk about 15ths as extensions is Jacob Collier, and I don't really feel qualified to talk about the musical uses of chords like C13(♯9♯11♯15). However, I did find a website talking about "augmented 15th chords", so it's definitely an extremely obscure - yet apparently legitimate - add-on to the regular system. Any legitimate 15th chord would have 8 notes in it, so obviously no 15th chord will ever be diatonic to any major key or its modes, but it could have its uses (Impressionism's one area I've heard the idea discussed).
When naming any chord, we only want one note from each scale degree, so instead of
G B D F A F#, we would spell out your chord as
G B D F# A E#.
Since E# is enharmonic to F, this gives us something like Gmaj9add#13 to notate the augmented 6th/13th when rearranged as
G B D F# A E#.
But this is awkward, as I don't see #13s in online chord charts, so we may prefer to note that extended chords have multiple possible root interpretations, so rearranging to
B D F# A E# G we now have a Bm7#11b13(no 9). Rearranging again to
D F# A E# G B we now have DM13(#9 no7), etc.