As you can see from here, Alistair Cook’s 244* at Melbourne is not his highest Test score.
That is 294.
However, it set a number of records. He became the first Test player to make 244*, while Bradman is the only one to be dismissed on 244 (vs England, who else?) in 1934.
Cook’s 244* is also the highest score by anyone carrying his bat through a Test:
The previous record was 223* by GM Turner vs WI in 1971-72; this series had all 5 Tests drawn and is the only such series not involving India.
The previous record for England was 202* by L Hutton vs WI in 1950-this was not enough to prevent an innings defeat against Ramadhin, Valentine and Goddard.
And the previous record in Australia was 169* by MA Taylor vs SA at Adelaide in 1998.
There was also a near-miss in 1974 when DL Amiss made 262* against WI in a total of 432/9.