Similar to how you can fill a whole 4/4 bar with a single whole note, what is the best or most correct/readable way to fill a 9/8 bar with one note? I was thinking about a dotted half note tied with a dotted quarter note, but it looks a bit funny.
Yes, it looks a bit funny, but you're right, the correct way to write a note that lasts for a full 9/8 bar is to tie a dotted minim (dotted half-note) to a dotted crotchet (dotted quarter-note). However, it is not correct to tie three dotted-crotchets or to tie a dotted-crotchet to a dotted-minim (with the dotted-crotchet first).
There are instances where you might use an unconventional grouping for a note lasting a full bar, such as this:
However, this is a special case; you wouldn't use a grouping such as this unless necessary.
Your solution is right, but in the end, it depends on how you phrase the bars in the piece. If for instance in a piano piece there's a rhythm
3/8 1/4 1/2 against it in the other hand, I would be tempted to use two ties and make the long note
3/8 ~ 1/4 ~ 1/2 as well.
Other than that, what Bob expains is correct: The standard notation is
3/4 ~ 3/8.
OK, but what then about the case where you want a note that fills a 9/8 bar, but there's a fermata on the downbeat? In your example of dotted half tied to dotted quarter, where do you put the fermata? On the dotted half? That makes it seem like there is then going to be an ictus on the dotted quarter. On the dotted quarter, you say? Well, that's clearly wrong, because we want the fermata on the downbeat.
This makes clear that we need a way to represent a note value that fills a bar, no matter what the meter signature is of the bar. I think the answer is: a whole note. In a non-4/4 or 2/2 context, the whole note should be understood to represent the entire bar.