A lot of programming language when you use an OR, he will check the first condition to be true and won't check the rest of the condition.
true OR (false) OR (1) OR (false) OR ....... OR (0)
with the first true it's enough to keep going and there is no need to check the rest. If we apply that theory to the LIKE.
Select * from T1 where col1 like '%me' or like 'me%'
would be a little bit faster than
Select * from T1 where col1 like '%me%'
because when he found some text with the text '%me' the condition will be true, it will show the result and keep searching to the next row.
What do you think about this?
EDIT: COL1 IS A VARCHAR(3) to have the same result (thanks Aaron Bertrand and Justin Cave for this recommendation)