This one was in the New York Times a while back:
2, 3, 3, 5, 10, 13, 39, 43, 172, 177, ... Give up?

0, 0, 0, 0, 4, 9, 5, 1, 1, 0, 55, ...
Hint: write out
one, two, three, four, five, six, seven, eight, nine, ten, eleven, ...
and think like a Roman! Give up?

From Chess Life, said to be a sequence that the world chess champion did not guess (except we would bet he was not given 10 terms):
7, 9, 40, 74, 1526, 5436, 2323240, 29548570, 5397414549030, 873117986721660, ... Give up?

Everyone knows about the even numbers, sequence A005843.
Less well-known are the eban numbers (the name is a strong hint!):
2, 4, 6, 30, 32, 34, 36, 40, 42, 44, 46, ... Give up?

Somewhat in the same spirit are the emirps:
13, 17, 31, 37, 71, 73, 79, 97, 107, 113, ... Give up?

Elegant, classic:
1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, ... Give up?

What about this one?
1, 3, 7, 12, 18, 26, 35, 45, 56, 69, 83, ... Give up?

Hmmm! Still thinking? I've seen nine-year-olds guess it quicker than that.

This one is easy for smart seventeen-year-olds, or if you've seen it before, otherwise not!
2, 12, 1112, 3112, 132112, 1113122112, 311311222112, 13211321322112, 1113122113121113222112, 31131122211311123113322112, ... Give up?

John Conway's astonishing analysis of the asymptotic properties of the above sequence is well worth reading - see the reference given.

Very easy:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25... Give up?

Also not so hard if approached in the right way:
1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, ... Give up?

The "RATS" sequence!
1, 2, 4, 8, 16, 77, 145, 668, 1345, 6677, 13444, 55778, ... Give up?