What are the major difficulties to the production of elements?

To get the information you wish, you'll need to dig a little deeper, as follows.

For any element in the periodic table, it is possible to calculate the quantity of work required to squeeze its constituent protons and neutrons together hard enough to stick. This is called the nuclear binding energy and is analogous to the heat of formation of a chemical compound. How to calculate this?

Knowing the individual masses of the protons, neutrons and electrons in any element, you can sum these together and compare the sum with the actual atomic mass of the element. For elements lighter than iron, there is a net energy release when the protons and neutrons fuse together and the actual mass is less than the sum. That mass difference is called the mass defect which is equal (via E = mc^2) to the energy released when the protons and neutrons fuse themselves together. For elements heavier than iron, the actual mass is greater than the sum which means net positive work must be done to create that nucleus (i.e., the nucleus contains stored potential energy, again equal via E=mc^2 to the mass difference).

In your example, where you wish to manufacture rare elements by squeezing other common elements together, E= (delta m)c^2 is the number you want. However, note the following:

This process can be carried out in a particle accelerator, where lighter particles are shot at heavier nuclei to produce still-heavier nuclei. But this is hugely inefficient because 1) the energy cost to speed up the "bullets" is enormous and 2) the vast majority of the lighter bullets either miss their target nuclei entirely, bounce off without sticking, or actually shatter the target to pieces.

That gross inefficiency makes it completely infeasible to build up rare elements in the lab compared to the cost of digging holes in the earth's crust, hauling ore to the surface, and purifying it to get those same elements.