The problem that got me thinking. $\left(\dfrac{\sin x}{x}\right)^2$
The overall pattern of this function is a parabola but it has been pulled down by $1/x$ and it wobbles because of $\sin x$.
So, in general is there any mathematical method/way to determine which function is the most dominate?
so for any $\left(f\left(g\right)\right)^{h}$ how could I tell which function $f,g,h$ will affect the shape of the graph most?
Your problem is an ill-posed one.
Note that there are many properties of graphs that you can call "shape" (for example regularity, it is clearly something different than the large scale geometry of the graph). Even if you choose one of the shape properties you have to specify how you measure its changes.
Last but not least: if you pick a function $\psi$ it is very unlikely that it has a unique representation $\psi=(f\circ g)^h$...