So, the question should say it all, but for example:
The G minor scale (G - A - Bb - C - D - Eb - F - G)
To change this into the G harmonic minor scale, I would need to raise the seventh note a half-step; (thus changing F to F#) though, that would mean the scale contains two flats and a sharp..
Would the scale (G - A - Bb - C - D - Eb - F# - G) be incorrect? If so, how can I fix it?
The G minor key signature is written with two flats, but the scale, as you noted, has a sharp in it. The G melodic minor scale has E natural and F sharp going up, and F natural and E flat going down.
The key signature is one thing; the scale is another. Some folk music uses scales very different from the major and minor we are used to.
I have seen music that used non-standard key signatures, like two sharps and a flat. It's rare, and is sometimes used when transcribing folk music.
Yes the minor scale has a raised leading Tone that can easily be a sharp. This happens in the Minor scales a few times. There is also the Whole Tone scale that can have sharps and flats. There is no rule that excludes them from being in the same scale.
The music notation will only enable permutations of the major scale (ie minor natural & all modes) to be written with only sharps or flats, because its structure is built-in the name of the notes (imagine what would happen if the notes were named from 1 to 12).
So yes, there are a number of old or synthetic scales that will need a special alteration scheme to be written, and the first two that will come to mind are, as mentioned in other answers, the flamboyant harmonic minor and the suave melodic minor.
Sticking with diatonic harmony, the most common examples are going to be scales that outline some extended harmony because there can be multiple altered chord factors which are either nonsensical or impossible to spell without a sharp and flat due to their function relative to the key. Pick up any Aebersold book and head to the scale syllabus and you'll find some examples.
Lydian dominant is a nice self-explanatory one - contains a ♯4 and ♭7. Diminished scales are also frequently impossible to write without a sharp and flat. Say we're in Cmin and want to play a diminished scale over the ii. ii°7 is D-F-A♭-C♭. Something needs to give in order to fill in the gaps without an augmented unison, so this scale will contain a C♯ as the 7th step in order to have an interval of a second between each step while still outlining the correct diatonic chord. Both half-whole and whole-half variants suffer this.
It's worth noting most of the Jazz scales commonly named after modes aren't necessarily used in a "modal" way. For instance the function of the aforementioned lydian dominant would be more likely found played over a V+11 than to actually emphasize the mode itself. The scales just share some common characteristics and inherit the name. Super-locrian is a notorious example that has almost nothing to do with the locrian mode other than it's underlying interval pattern.
