At the outset, we have:

11^2003 times 7^2004 times 13^2005
= (11*7*13)^2003 times 7 times 13^2
= (1001)^2003 times (7*13) times 13
= (1001)^2003 times 91 times 13

Now,we observe that:

The last digit of 1001^2003 is 1
The last digit of 91 is 1
The last digit of 13 is 3

Consequently, the required last digit of the given expression
= 1*1*3
= 3