Then probability for having consecutive k Heads is in your case
0.5 * (1 + 0.5 ^ k),
so k + 1-th is just
0.5 * (1 + 0.5 ^ (k + 1))
So for k = 3 it's 0.5 * (1 + 1/8) = 9/16
For k = 4 it's 0.5 * (1 + 1/16) = 17/32
As you can see, it steadily goes down, for very high number of flips it will get very close to 0.5, but still marginally above 0.5. The limit for this formula in sense of k -> Infinity is of course 0.5, but you can't flip infinitely many times.
EDIT: I think that I have messed up, but I can't put my finger on it. If it's any excuse, I was never any good with probability. How does one differentiate consecutive case from consecutive and a next flip? This smells of Markov Chain to me, but that's just a hunch.