The Multi-Moon Orbiter concept is introduced, wherein a single spacecraft orbits several moons of Jupiter, allowing long duration observations. The Delta-V requirements for this mission can be low if ballistic captures and resonant gravity assists by Jupiter's moons are used. For example, using only 22 m/s, a spacecraft initially injected in a Jovian orbit can be directed into a capture orbit around Europa, orbiting both Callisto and Ganymede enroute. The time of flight for this preliminary trajectory is four years, but may be reduced by finding a compromise between fuel and time optimization during the inter-moon transfer phases.