Confused by the definition of Linear Complexity of an LFSR sequence?
Length 2: one tap at the rightmost cell, going back into the left side without further additions. Starting with initial filling $01$ we get the sequence $101010\ldots$. The same sequence can also be achieved with two taps on both cells and the xor of them going in the left and the same initial filling. With register length $1$ only constant sequences are possible in Rueppel's definition: I don't think he even really considers such trivial cases.

So (10)* has linear complexity 2.