I've found some sources stating that for two natural transformations $\alpha, \beta: F \rightarrow G$, no non-trivial modifications can be found in standard categories. However, it is possible to define natural transformations as functors $\alpha, \beta: \mathcal{C} \rightarrow \mathcal{D}^{\rightarrow}$ if $F,G:\mathcal{C} \rightarrow \mathcal{D}$. So why don't we investigate functors $\mathcal{C} \rightarrow (\mathcal{D}^{\rightarrow})^{\rightarrow}$, and treat them as natural transformations between natural transformations (seen as functors)?
If I understand correctly, objects in $(\mathcal{D}^{\rightarrow})^{\rightarrow}$ correspond to commuting squares in $\mathcal{D}$ and arrows to commuting cubes. Is that correct? If yes, then this gives rise to four functors $F,F',G,G':\mathcal{C}\rightarrow\mathcal{D}$ with two natural transformations $\alpha:F\rightarrow G, \beta:F'\rightarrow G'$ and a "modification" $\alpha \rightarrow \beta$ with components being the edges of the cube that connect the naturality squares of $\alpha$ and $\beta$.
Does this lead to non-trivial structures? Also, how does this correspond to the idea of modifications in higher category theory?
"Modification" is actually a technical term from 3-category theory (nLab reference).
As an organizational philosophy, I think the better perspective on natural transformations is that the functor category $\mathcal{D}^\mathcal{C}$ is a fundamental notion, and that natural transformations are simply the arrows of $\mathcal{D}^\mathcal{C}$.
Since $\mathcal{D}^\mathcal{C}$ is simply a category, it doesn't have (nontrivial) 2-arrows between its arrows.
Note that assuming that functors form the objects of an exponential $\mathcal{D}^\mathcal{C}$, we can determine the arrows by the product-exponential adjunction.
Let $\mathbf{2}$ be the arrow category $\bullet \to \bullet$. There are (natural) bijections
$$ \mathrm{Arr}(\mathcal{D}^\mathcal{C}) \cong \mathrm{Funct}(\mathbf{2}, \mathcal{D}^\mathcal{C}) \cong \mathrm{Funct}(\mathbf{2} \times \mathcal{C}, \mathcal{D}) \cong \mathrm{Funct}(\mathcal{C}, \mathcal{D}^\mathbf{2}) $$
By the third term, we see that natural transformations can be expressed as functors $\mathbf{2} \times \mathcal{C} \to \mathcal{D}$. Your description of them appears in the fourth term.