For those who aren't aware, Four Fours is a puzzle where the objective is to express as many consecutive integers as possible, starting from 1, using a finite number of allowed operations and (as the name suggests) four 4s.
Now, a Discord server I am in is doing this puzzle collectively. To make things easier we have tried to devise a list of possible numbers that can be created from a single $4$ and a series of permitted operations applied to it, so that the problem then becomes "create [insert number] by adding, multiplying, subtracting and dividing these numbers".
During the process of this, one person created a conjecture: perhaps it is possible to create all positive using a single $4$ with a finite number of floor, gamma and square root function applied to it. For example, one could write
$$\left\lfloor\sqrt{\sqrt{\Gamma\left(\sqrt{\Gamma(\Gamma(4))}\right)}}\right\rfloor=42$$
And thus you can use a combination floor, gamma and square root functions applied to a $4$ to get $42$.
It may be worth mentioning that somebody managed to check that it is indeed possible to get all numbers between $1$ and $142$ through this method, but couldn't check any further due to overflow issues.
(Note: similar to this question, but we allow the floor function here)
