a continuing series for the gravitationally impaired
The best orbit is the minimum energy orbit. It's just above the atmosphere at about 100 miles high, and precisely circular. Today it is called a parking orbit, where payloads are parked until they are checked out before being boosted to higher orbit. If something goes wrong with the satellite, atmospheric drag will soon bring the payload back into the atmosphere, safely destroying it. It takes a very precision launch vehicle to hit that minimum energy orbit. Hitting the minimum energy orbit is also a great demonstration of your precision guidance system. Precisely hitting an orbit is a good demonstration of your country's military ability to precisely hit a target with an ICBM. Precision guidance requires a precision inertial system for out of sight control or precision tracking radars and down range tracking stations to control the vehicle, all very expensive. The Soviets demonstrated this ability, the US demonstrated this ability, and now no one questions the need to hit the minimum energy orbit for a satellite launch. Except me of course.
Getting something into orbit cheap is my goal. A precision guidance system is very expensive and subject to strategic weapons restrictions. Also I'm not on the list of people who can own an IMU. So, can you get into orbit with a really cheap guidance system, from one location. MAYBE. Go straight up, like a cheap sounding rocket, way way up, like maybe 115,878 miles up. It takes a lot of energy, more than the minimum energy orbit, but the guidance problem is easy, the vehicle is always in sight, straight over head, well at least during the boost phase. You can use lots of cheap dumb stages and just keep pushing straight up. Now, the fun part is what you do when you get way up there, you turn left. :-) The idea is to hit a highly elliptical orbit with the apogee way out there and the perigee (the close approach to earth) just above the atmosphere. A satellite in a highly elliptical orbit will have very low velocity at apogee and very high velocity at perigee. A sounding rocket coasting out to a very high altitude apogee on a straight up flight path would of course come straight back down. Give a little kick, not much really, 487 ft/sec, to the side, at apogee, and the descent would miss planet earth and go into orbit. Whatever the case, you would want to reach apogee a multiple of 24 hours after the launch, as in the 48 hour case above, then the orbital insertion event would occur in line of sight, directly overhead from your launch facility, very easy to control, cheap. John Eisele gives a good discussion of such an orbit in his book "Astrodynamics, Rockets, Satellite, and Space Travel", well, he discusses actually hitting planet Earth, but he also defines the criteria for missing the planet too. In case you're wondering, you can't get into orbit with a straight sounding rocket, because the point of engine cutoff is always a point on the orbit and sounding rockets finish too low in the atmosphere. The straight up shot would have a safety advantage in that failure to insert into orbit correctly would result in the satellite destroying itself reentering the atmosphere, some multiple of 24 hours later, which also suggests a reentry and recovery possibility but that would need better guidance again. High apogee insertion would also allow access to polar orbits as easily as an orbit with the plane at the launch site lattitude, or you could lean south and try for equitorial. While this is probably not a good launch scenario for big commercial payloads, it can be used for a first time launch, by an entrepreneurial launch group, trying to do a launch on the cheap.
P.S. If you're developing nearly escape velocity anyway, why not go to the Moon, it's actually easier than getting into orbit, no left turn, just launch towards the Sun three days before new Moon, when you get there the Moon will be there and the sphere of influence of the Moon is about 60,000 mile in diameter. Add a little retro fire to slow down and the payload should drop onto the Moon.
There is another method of getting into orbit with a sounding rocket, It was used by the Pilot or NOTSNIK program.
You launch a spin stabilized satellite at a 50 degree angle in to an elliptical trajectory, something like an ICBM trajectory (scarry!). As the satellite is just about to reenter the atmosphere a retro rocket is fired to change the trajectory into a low earth orbit. The orientation of the satellite is such that at retro fire the vector addition of delta vee is a circular flight path. The method requires precise launch angle, launch velocity, and retro timing. It has never been sucessful.
A good description of the technique
Another good description of the technique