Yes the answer is 1.
Sagar I have a different and simple approach.
Here comes our next basic funda.
All of us know that the remainder obtained when a number is divided by10 is nothing but the unit's digit of that no.
Had u ever thought of the remainder that is obtained when divided by 9. It is nothing but the single digit sum of that number.
For example the single digit sum of 4658 = 4 + 6 + 5 + 8 = 23 = 2 + 3 =5.
Now when 4658 is divided by 9 the remainder is nothing but the remainder obtained when sum of the digits of 4658 i.e 23 is divided by 9 which in turn is nothing but the remainder obtained when 2+3 = 5 is divided by 9.
So that's the funda here folks.
Finally the remainder obtained when 4444^3333 is divided by 9 is nothing but the single digit sum.
4444^3333 % 9
= 7^ 3333 % 9
= 343^ 111 % 9
Since the remainder obtained when N is divided by 9 is 1 the single digit sum is 1.
Remember if the remainder obtained is 0 then the single digit sum is not 0 but ofcourse it is 9