Ok, I feel a little bit ashamed by my question.

This afternoon in the train, I looked for a counter-example:
— k a field
— A a finitely generated k-algebra
— B a sub-k-algebra of A that is not finitely generated

Finally, I have found this:
— k any field
— A=k[x,y]
— B=k[xy, x.y^2, x.y^3, ...]

(proof : exercise)

My questions are :

1. What is your usual counter-example ?
   
2. Under which conditions can we conclude that B is f.g. ?
   
3. How would you interpret geometrically this counter-example ?