The main concern with eating bananas is constipation. Bananas are extremely sweet, and that can be somewhat dehydrating (which is the leading cause of constipation). If they're eaten in an unripe state (a ripe banana is yellow with some brown speckles), they may also be harder to digest, as their starch content is higher.
However, how that affects your daughter will depend on, well, your daughter. If she tends to hydrate well, and her bowels work pretty well, then she'll be fine - just pay attention to how her stools are 8-24 hours after she eats the bananas, and if it's a problem, cut back in the future.
When she potty trains (if she hasn't already), you may want to avoid constipation more than usual for a while (as it can make it hard on the child when trying to go), but again that's going to be on a person by person basis.
Finally, I would make sure she's eating enough other foods. As long as she is getting plenty of protein, green veggies, dairy/calcium, and the other things she needs, and the bananas aren't causing problems with her digestive tract, then I'd say go ahead and let her eat what she wants. As Brusselssprout notes above, letting her pick how much to eat is the healthier way to go, so long as you give her a selection of healthy foods and make sure this isn't replacing her full meal.
If you do find it is replacing her full meal, then treat it like dessert or snack - only give it at times a few hours before the next meal.