The first (upper) qubit is the one we want to teleport, so could start it any state, call $\alpha|0\rangle+\beta|1\rangle$. Our goal is to teleport it to the third (bottom) bit.
After entangling the second and third bits, we have
After applying CNOT from the first to second bit, we have
At this point, we measure the second qubit. If it's $|0\rangle$, then we end up in the state
If it's $|1\rangle$, then we end up in
In this case, we apply X to the third bit, yielding
In either case, if we have access to the third qubit only, it appears to be in the state $\alpha|0\rangle+\beta|1\rangle$, which is what we want. What's wrong with this scheme? The only thing I can think of is that it "doesn't count" as teleportation because the first and third qubits are still entangled at the end. But I thought the main point of teleportation was just to transmit a state using only classical data and a pre-entangled pair, in which case this does seem to "count".