The number of moons
Sean Raymond wrote a great answer to a similar question, and I'd like to base my answer on his. He used the following technique to estimate the total number of moons orbiting a planet, assuming all the moons are in mean motion orbital resonances with one another.
- Calculate the outer edge of the region of stable orbits, which is one half of the Hill radius. For Earth, this distance is $\sim0.005\text{ AU}$. For comparison, the Moon's semi-major axis is $\sim0.0025\text{ AU}$.
- Estimate the inner edge of the region. I'll guess that it's 1/20th of the Moon's semi-major axis, or $\sim0.000125\text{ AU}$. The very inner limit is the Roche limit. However, that depends on the mass of the orbiting body. Without knowing what the moons would be like, I'm not inclined to speculate on what the Roche limit would be (we're probably safe, though, with this as our inner bound).
- Divide the outer radius by the inner one. That gets you $40$.
- Kepler's third law relates semi-major axis $a$ to period $P$ by
$$a^3\propto P^2$$
Raise your previous result to the $3/2$ power. This gets us a factor of $252$, which is close to $256=2^8$. This implies that we could fit $8+1=9$ moons comfortably in orbit, if we bend the limits a bit.
Some extra thoughts on stability:
- The masses of the moons would need to be adjusted so as to not perturb one another too much. I'd guess they would all have very little mass, but I don't know for sure. It's actually an interesting question, come to think of it: how small can you make a moon that will clear its orbit in a debris disk? I'll update this if I find a good answer.
- The formation of the system would be difficult to explain. Stable resonances can of course arise - and we do see this in the Jovian and Saturnian moon systems, specifically, some of the Galilean moons - but this many moons would be a bit much. You'd also need a rather large debris disk around Earth. I don't think the Giant Impact Hypothesis could work here!
- The system would likely be stable if it was isolated, but if an outside body came by, things would go wrong rather rapidly.
The effects
Whenever you create a system of moons, tides tend to come to mind. The site has covered the tidal patterns from multiple moons in some depth (see How would having multiple moons affect tides?), although never with 9 moons, I believe. The method to model tides would be to simple treat the cycles as the sum of 9 sine waves, one for each moon, and look at the patterns that arise.
Culture is always an interesting one to think about. Many civilizations had deities for the Sun and the Moon; maybe you'd have something similar here. 9 moon deities give you fodder for plenty of interesting stories. For me, the myths centered on the Pleiades comes to mind. Calendars might also be different; more complicated timekeeping systems would certainly be possible. You might want to explore that angle a little.
