Let $C>0$ be given and call a function $f:\mathbb{R}^N\to\mathbb{R}$ basic if it contains only addition, multiplication or exponentiation. So e.g.
$$f_1(x_1,\dots,x_N) = \sum_{i=1}^N x_i$$
$$f_2(x_1,\dots,x_N) = \prod_{i=1}^Nx_i$$
and if $g(x,y) = x^y$ for $x,y>0$, then
$$f_3(x_1,\dots,x_N) = g(x_1,g(x_2,\dots,g(x_{N-1},x_N)\dots)$$
are all basic functions. Furthermore, call a partition of $C$ a list of $N$ numbers $(t_1,\dots,t_N)$ for an arbitrary $N\in\mathbb{N}$ such that 1.) $t_i > 0,i=1,\dots,N$ and 2.) $\sum_{i=1}^N t_i = C$. So e.g. $(C)$, $(C/2, C/2)$, $(C/3, C/3, C/3)$ are all valid partitions for $C$.
Given the above, I was wondering how me might want to choose both the partition $(t_1,\dots,t_N)$ and the basic function $f$ such that $f(t_1,\dots,t_N)$ is as large as possible and furthermore how we might quantify this. That is, optimize 1.) total number of numbers $N\in\mathbb{N}$, 2.) the partition $(t_1,\dots,t_N)$ itself and 3.) the basic function $f$.
As of writing my only idea/observation/conjecture is that if $C > 1$ is sufficiently large enough, we want to take as many consecutive powers $x_1^{(x_2^{\cdots}}$ as possible (i.e. something of the form of $f_3$) with $x_1 > 1$ and $x_2,\dots,x_N$ large, but not too large enough. An example, if $C = 10$ and we were to obey by a simple rule that we try to take as many equal $x_i$s as possible, with any leftover set to e.g. the last of the $x_i$s, then $3^{3^{4}} = 4.4342649e+38$ but $2^{2^{2^{2^{2}}}} = 2^{65536}$, which is quite big.
So taking many $2$s seems to be better than taking $3$s. But are $2$s optimal? Could there be some $0 < \varepsilon < 1$ such for $t_i = 1 + \varepsilon$ with sufficiently many $i$s, there exists a basic function $f_\varepsilon$ such that $f_\varepsilon(1+\varepsilon, 1+\varepsilon,\dots,1+\varepsilon,t_{k(\varepsilon)},\dots,t_N) > h(2,2,\dots,2,t_{k(2)},\dots,t_N)$ for every basic function $h$ such that the partition $(t_1,\dots,t_N)$ contains as many $2$s as possible? As of writing I can't really say anything. This seems like a fun game/puzzle, but I think that either some conclusive result already exists or we are going to need quite a many lemmas to quantify this problem better. What do you think? Do you know more about this or have you encountered such "optimization" problems before?