As Rick alluded to above some 3 and 4 engine aircraft have provisions for this sort of thing. I can only speak to the C-130. There were provisions for three engine take-off's in the -1. Min runway length, OAT, gross weight...etc... all within some pretty tight tolerances. In addition the MAC put even further restrictions on it to the experience of the crew I believe. I suspect that it required MAJCOM approval to perform anyway. We also had a provision for doing a "windmill taxi start" in the case of a blown starter on one of the engines. You had to have 7 or 8,000 feet of runway and you accelerated down the runway using the windmilling prop to start the engine. You then taxied back and took off. Again.. lots of restrictions etc... but it was an authorized procedure.