It depends. There are trees that grow very fast, but these trees tend to be large and likely to have broken limbs. Examples include silver maple (Acer saccharinum), cottonwood (Populus deltoides) and pussy willow (Salix babylonica). Their roots can also cause damage to infrastructure (sidewalks, foundations, etc) because of their size. They are not inherently bad, their characteristics just need to be taken into account.
There are also trees that grow fast yet aren't as large and are safer like eastern redbud (Cercis canadensis) or river birch (betula nigra).
I'm not aware of trees that reach a maximum size in 5 years. But if you tell me more about where the tree is going to planted, and also if there is a maximum size for the tree, then I can give additional recommendations. I'm sure there is a species for the job!