what must a monarch achieve to be referred to as "the Great" ?

There is nothing in particular that someone must achieve.

Usually, "the Great" was appended to someone's name because he or she did something very striking in terms of conquest or utterly altering the character of the nation socially or politically or intellectually.

However, we don't use such epithets today and that one has been used for only one monarch in the UK -- the medieval King Alfred, who was the precursor to a unified English monarchy. If Elizabeth I wasn't called "the Great," Elizabeth II won't be. She is, after all, only a constitutional monarch without the powers, potential, and possibilities for altering her nation that her much earlier predecessors had.

No ..adding those type of suffixes after a monarchs name fell out of practice many centuries ago .they were used to differentiate between Monarchs with the same name.Now they just have the Roman numerals to show which George, Edward etc that they were/are

In the past a Monarch who was known as Great would probably have been remembered as a great military leader (The names were given many years after the Monarchs death )

In the modern times, adding 'the great' is impossible. That's so ancient. That's why world monarchs uses Roman numerals after their regnal name.

I think mainly anyone with 'the Great' gave themselves that title, e.g. Alexander the Great.