One possible solution would be to turn the highest mountain into a space gun. Depending on the density of the atmosphere at that altitude, either go straight to orbit or launch a vehicle (rocket) into a low orbit and from there use propellant to get free of the gravitational well.
If we use the 9680km radius from the article and assume average density equal to earth we end up with a planet that has the following characteristics:
Earth Avg Mass: $$5.98 * 10^{24}kg / (1.33 * π * 6,378,000m^3) = 5516 kg/m^3$$ Our Planet’s Mass: $$(1.33 * π * 9,680,000m^3) * 5516 kg/m^3 = 20.9 *10 ^ {24} kg$$
Our Planet’s escape Velocity: $$( 2 * (6.67 * 10^{-11}) * (20.9 *10^{24} kg) / ( 9.680 * 10^6 ) )^{0.5} = 16.9 km/s$$
Surface gravity: $$(6.67 * 10^{-11}) * (20.9 *10^{24}) / ( 9.680 * 10^6 )^2 = 14.88 m/s^2$$
Given the increased gravity we are unlikely to see mountains as high as the ones on Earth, but let's ballpark it and assume a 5km peak maximum.
On earth we have drilled as deep as 12 km into the crust (granted with a 2 inch bit), so it is not out of the question that in the near future we will be able to dig a tunnel from the top of the mountain and continue 5km into the crust, basically creating a 10 km long barrel.
We have rockets, even tiny ones like the SS-520-5 which can achieve orbit from earth. So to prove the concept we could have the space gun place a rocket at an altitude where the escape velocity equals that of earths surface. $$ \sqrt((6.67 * 10^{-11}) * (20.9 *10^{24}) / 11.2 m/s^2) = 11,156,476m $$
Our space gun must be capable of placing the rocket at: $$11,156,476 – 9,680,000 = 1,476km $$ above the planet surface. 921
Well that’s not going to work, since the furthest we have ever gotten with a space gun like setup is sending 180kg up 180km.
Second option would be to accelerate the rocket so it leaves the mouth of the barrel with enough velocity to make up for the increase in escape velocity. $$16.9 km/s – 11.2km/s = 5.7km/s$$ Ian McNab proposed a design for a railgun that could accelerate a 400kg projectile to 7.5km/s back in 2003.
However, I must admit the math is getting away from me when I try to build a model that accounts for atmospheric density at 5km altitude on our imaginary planet. The drag is enormous and there are huge structural challenges in accelerating an existing rocket to the tune of 165 Gees and not have it burn up once it leaves the mouth of the gun. In the end a coherent solution was beyond my ability. 234