I have successfully started moving in 2nd or 3rd gear on a manual trasmission vehicle. 2nd gear is reasonably easy to do, you just need to release the clutch later. 3rd gear will smoke-test your clutch.
The reason you don't do this often is that the engine has a lowest possible RPM at which it produces useful torque.
I disagree with the upvoted answer that the engine doesn't have enough torque. Newton's 2nd law says F = ma, i.e. force is mass times acceleration. If you can accelerate in 5th gear from 100 km/h to 120 km/h (ok, it's slow but you can do it), you can accelerate in 5th gear from 0 km/h to 20 km/h using the same force/torque in the same time.
The problem is that the lowest possible RPM at which the engine produces useful torque is limited to something like 1000 RPM. So, accelerating in 5th gear from 0 km/h to 20 km/h slowly would totally kill your clutch. You'd have to use the clutch the entire time of the slow acceleration.
If you are driving on totally flat and smooth ice without studded tires, it actually might make sense to start on another gear than the 1st gear. The torque of the 1st gear is just excessive for the situation.