Curve Fitting By Predict polynomial degree with ANNs
Interesting idea :)

The things I would consider:

Problem definition

Should it be classification or regression? If you define it as classification, you should select the number of classes, hence the maximum and minimum polynomial degree (e.g. 1,2,3,4,5,6,7,8,9 and 10 could be polynomial degrees which you recognize).

If you select regression, you don't need to define the maximum number of classes, but the output layers could be trickier to define.

Dataset

How do you define your samples? I guess that it's easy to generate samples (select some polynomial with degree $n$, sample points from it and then all those points define one sample for class (or regression value) n).

It would be interesting to create the image out of the sample (plot of points) and then maybe use a convolutional network to learn the classification (or regression).

Also, consider how dataset big should be. It's hard to say, you will have to test it and see how big it should be.

Losses

For classification probably cross entropy and for regression probably mean squared error.

Let me know the results.