In 'From the Earth to the Moon' (1873) by Jules Verne, A huge cannon is used to send a spaceship to the moon. A lively discussion in chapter IX leads to using 400,000 pounds of fulminating cotton to launch their ship to the moon.
Two questions;
The Delta V requirement to launch is about 14 km/s to low lunar orbit, per Wikipedia. That means that you would have to achieve a speed of 14 km/s in order to orbit the moon. Some of that will need to be done from space, but most of it could theoretically be achieved from the ground. So, what do you need to do to make that happen?
In World War II, the Germans developed an artillery shell that could travel at 1.67 km/s. It used 200 kg of powder, and fired a 106 kg shell. Let's just suppose that you could scale that upward infinitely (Not likely, but we'll just assume for a moment). Furthermore, let's assume a mass of 1000 kg for the ship (Likely would be higher). Given all of that, you would need 10 times as much to launch the ship the same speed, and about 72 times as much to launch the ship to lunar orbit. That would scale to about 14400 kg of powder, or about 16 tons of powder, much less than Jules Vern stated you would need. So, why don't we do that?
While in theory one could get to the moon like this, pure cannon thrust would not be sufficient to land on the moon, at least in a controlled manner. You would end up landing on the moon at the lunar escape velocity of 2.4 km/s, without a rocket to stop you. Furthermore, the gravity forces exerted on you would at launch would be enormous, artillery shell electronics have to be rated at 15000 Gs. Good luck getting a person to survive that. And also, the physics doesn't quite scale as I indicated here, but the numbers provide a good first order approximation.
The launch profile of a rocket is near best case for getting astronauts to Space, in terms of the amount of gravity. You really do need to be continually thrusting for some time. However, a rail gun could provide some of the velocity needed to orbit, if you plan carefully to make this happen.
As far as I know, the shock wave in detonating explosives does not go faster than about 2.5 km/s, so a bullet will not be propelled beyond that speed, however many barrels of gunpowder you accumulate. The shock wave can be sped up if the operation occurs in a high pressure environment, though, but reaching enough speed to get to orbit (about 8 km/s) seems hard, let alone going to the Moon.
You can make a multi-stage system, though: one cannon which shoots another cannon which shoots a third cannon, and so on. Ultimately you end up with a rocket, not a cannon. It is a bit of a question of definition...
(A variant with an explosive a bit more punchy than gunpowder is being seriously investigated, but I doubt it will happen any time soon.)
The acceleration would also flatten in a very literal way any vertebrate unlucky enough to have been selected for this trip. Space cannons are more than idle speculation, but they are for launching bulk materials, not people.
The absolute theoretical maximum mass that 1,600,000 lbs of powder could launch to the moon is a bit under 35,500 kg. The calculation is not all that difficult (introductory calculus-based physics), but is somewhat long and involved, and would be quite ugly on a site like this without mathjax. However, that makes two major assumptions that cannot actually happen: no air resistance, and all of the powder burns instantly, transferring all its energy to the rocket (none to a visual flame, none to sound, etc. I'm still working on a quantitative analysis of these effects, but I'm quite certain that accounting for either air resistance or the finite burn speed of the rocket would make it impossible to reach the moon.
The maximum speed a propellant-powered rocket can reach depends on the rocket's mass, the mass of propellant, and the exhaust velocity of the exhaust. As per James Jenkins, the ship was 20,000 lbs; the propellant is, of course, 1,600,000 lbs. Using a typical black powder exhaust velocity of 800 m/s. Without fighting gravity, that amount of gunpowder could propel the ship to a bit over 3,500 m/s, well short of the Earth's escape velocity of 11,200 m/s. Turning it around a bit, that mass ratio would require an exhaust velocity of nearly 2,550 m/s. And for completeness, the given 160M lbs of propellant could launch a rocket of a bit under 1.5 lbs; it would take 24 billion pounds of propellant to launch the full 20,000-lb rocket.
There is a story told that the first manmade object to achieve escape velocity, was a man hole style cover, over an exhaust vent, from an underground nuclear bomb test.
However, they quote this on the website.
But the assumption that it might have escaped from Earth is implausible (Dr. Brownlee's discretion in making a priority claim is well advised). Leaving aside whether such an extremely hypersonic unaerodynamic object could even survive passage through the lower atmosphere, it appears impossible for it to retain much of its initial velocity while passing through the atmosphere. A ground launched hypersonic projectile has the same problem with maintaining its velocity that an incoming meteor has. According to the American Meteor Society Fireball and Meteor FAQ meteors weighing less than 8 tonnes retain none of their cosmic velocity when passing through the atmosphere, they simply end up as a falling rock. Only objects weighing many times this mass retain a significant fraction of their velocity.
From another amusing perspective, there is a great science fiction story, called King Davids Spaceship by Jerry Pournelle that postulates a universe where the interstellar faring societies will not intervene unless your planet can reach orbit, and for political reasons a planet requires space flight as soon as possible, so they build a manned craft, that uses the approach of firing a gun downward, (sort of chemical version of Orion) to reach orbit.
Pournelle is fun for writing good science fiction, and it has interesting discussions of the issues involved.
