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 $k$-subalgebra of $A$ that is not finitely generated

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

(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 ?