I know the classic Four Fours puzzle, which asks to create whole numbers using exactly four "$4$"s combined via certain operations (arithmetic, concatenation, grouping, factorials, etc.), the exact roster of which can vary.
I have tried this specific version of it: You can use only the basic arithmetic operations ($\times \div + - $), square root ($\sqrt{}$), and concatenation (eg, "$44$" is allowed).
How far you can go? I cannot go further than $18$. I am stuck with $19$ and does not seem that there is a solution for it.
Let $\#$ denote concatenation. That is, $4\#3 = 43$. Then
$$(4 \# \sqrt 4) / \sqrt 4 - \sqrt4 = 42 / 2 - 2 = 19.$$
Not sure if you can use parentheses, but you would just have to perform concatenation first above all else.