The theory is here. It is pretty simple: form any integer bigger or equal that 0 using four fours and symbols.
Is there any demonstration which explains why with four fours is possible to form integers starting from zero? Which would be the maximum (not the maximum integer possible to form with four fours, the maximum integer until all its smaller integers can be formed)?
Did you read the article you linked completely? It says: "Paul Bourke credits Ben Rudiak-Gould with this description of how natural logarithms (ln()) can be used to represent any positive integer ''n'' as $n = -\sqrt4\frac{\ln\left[\left(\ln\underbrace{\sqrt{\sqrt{\cdots\sqrt4}}}_{n}\right) / \ln4\right]}{\ln{4}}$."
I presume this answers your questions.