What situation would make a launch window rare? What is the rarest known launch window?
The launch of New Horizons was very critical. The NASA scientists had 5 years to develop a space probe to Pluto and make sure it got a gravity assist through Jupiter, which is only possible if launched in the year 2005 to 2006: although they had 5 years of time to prepare and launch, they faced several problems fixing the Atlas 5 rocket, budget issues, government pressure and the most amazing part is that, if this window were missed it would take 11 years for it to reach its target Pluto, for which the fuel may not be sufficient, hence this was a race against time and a very interesting launch window. A similar possibility is 2 to 3 decades away.
Comet West with its unpredictable but estimated at 558 000 years period is a good contender.
Since launch windows can aim for the alignment of N objects, the answer obviously tend to infinity.
You also need to remember than launch windows are just "optimal launch times"
Using non Hohmann transfers, or long parking orbits, you can launch any time (using more delta v).
Hohmann launch windows occur each synodic period. Or a more general version of a Hohmann transfer would be a transfer orbit tangent to both departure and destination orbits. This also occurs each synodic period plus or minus.
Call period of departure orbit T1. Call orbital period of destination orbit T2.
Synodic period = |(T1*T2)/(T1-T2)|
So for example an near earth asteroid with a period of 1.1 years would have a Earth-asteroid synodic period of 11 years.
So the more earthlike an orbit is, the rarer the launch windows. As objects grow more distant, launch windows get closer to annual. For example the Earth-Neptune synodic period is 1.006 years.
Shoemaker and Helin's Earth Approaching Asteroids As Targets For Exploration is an oft cited paper for delta Vs it takes to reach an asteroid. Their delta V calculations call for rendezvous when asteroid is at aphelion. Which eliminates most of the tangent transfer orbits. This can drop launch windows to centuries apart.
If you're okay with the transfer orbit making several circuits about the sun before asteroid rendezvous, that makes possible launch windows more frequent. But that'd make for trip times on the order of years. Multi-year trip times are okay for sending machines to an asteroid but that makes sending humans a lot harder.
Near-optimal (or at least reasonably cheap) encounter between bodies in strongly eccentric, similar orbits (similar argument of periapsis, inclination, period).
For optimal transfer, one part of the transfer should be near periapsis of the lower orbit, the other near apoapsis of the higher orbit. That's fine; that's possible every period of the starting orbit. Except the body you want to encounter in the other orbit will be somewhere completely else when you're at the periapsis. It's a problem similar to provided by HopDavid, except you can't initiate your transfer whenever the craft at any point of the orbit is aligned with the target, so on top of body A being the correct angle ahead of body B, you need body A at periapsis and body B being the correct angle before its apoapsis - roughly squaring DavidHop's time estimate.
Interestingly, for orbits even of similar period but much more different, this may be easier, as other maneuvers to match other orbital parameters (inclination, argument of periapsis) take so much work fixing them, they leave a lot of wiggle room to squeeze matching the phase/orbital angle into these maneuvers.
This question has six answers already, but I would like to include a more general answer:
In solar system spaceflight there is a trade-off between fuel (delta-V) and mission duration. You can go almost anywhere you want with very little delta V providing you don't care how long it takes. Conversely, if you have an arbitrarily large delta V budget, you can go places very quickly.
At one end of the scale is the 'interplanetary network', a set of low-energy transfers that are constantly changing and evolving. By using tiny nudges from gravitational assists and passing through Lagrange points, you can go almost anywhere with little fuel, but journeys can take millennia to complete.
At the middle of the scale is Hohmann transfers, the sort of 'default' maneuver that represents the lowest energy transfer in a two-body scenario (i.e. when you don't have the gravitational influence of other bodies to nudge the craft in any given direction). Kerbal space program players will be familiar with this type of transfer.
At the other end of the scale is Pointing the nose of the craft at the planet and putting the pedal to the floor. This is only feasible for the sort of nuclear-powered torchships that humanity won't have access to in a long time. But with high delta V and a high acceleration, you have the luxury of taking much quicker but much less efficient trajectories towards your target.
So what does this have to do with launch windows? The same dichotomy applies. The higher your Delta-V budget, the less dependent you are on specific launch windows.
On a torchship with arbitrary delta-V, you can launch essentially whenever you like.
When using Hohmann transfers, as per HopDavid's answer, you have launch windows that depend on the relative difference between the orbital periods, which works out to just under ~1 a year for outer planetary objects.
When launching with a very low delta-V budget, the combination of assists can range from a rare occurence (as in the Voyager 2 grand tour) to a unique occurrence that will never happen again in precisely the same way.
It’s great to have a planned window. But one should also consider how important it is to have clear weather at the launch site at engine ignition. We used to launch shuttles and a wonderful launch director named Bob Sieck (who had education in meteorology). He famously used to be able to put the shuttle up in a weather “hole” From Kennedy Space Center. He had to always watch the weather and plan when we fueled to get to the hole at the right time. It was amazing! You can have all the plans you want... but mother nature and the health of your rocket ultimately control if you fly.