Can one get more than 3 suggested routes via Google Maps?
If so, how?
It doesn't appear that way, according to this unofficial blog post about the feature, which only mentions 3 suggested routes.
However, there is one method of getting a few more alternate routes, by using the Avoid highways or Avoid tolls options, which are visible when you click on Show options. More info about this feature.
I've noticed that if you have more than two points, it won't offer alternative suggested routes, so make sure you only have two points defined. Similarly, you should avoid dragging the route, as it also removes any suggested routes.
A workaround, since it looks like you can't quite do this in Google Maps automatically:
I would try breaking the route into two or three parts (depending on length), and then changing the intermediate point(s) a little bit. See what suggestions you get on the partial routes.
You can combine them into one route when you're done, but unfortunately, you'd need to do all this manually.
Another idea is to play with the different modes: Driving, walking, and cycling will often give you very different routing suggestions.