@Michael Hall said it best, in the first sentence of item 1 of his answer:
There are simply too many variables for a definitive answer.
Here are two extremes to illustrate the range of answers insofar as altitude is concerned:
Let's say you have a 747 at it's optimum altitude in cruise with all four engines operating, say 37,000 just for the heck of it. Now let's say you lose one engine. Can you now climb on three engines? No, in fact you won't even be able to keep the altitude you have. You will start drifting down.
Now let's say that same 747 with the same load (actually heavier because it would still have the fuel on board that it used to get to its 37,000 foot cruise) lost an engine while on the takeoff roll but after V1. Will it climb? Yes, even with the drag from the gear and takeoff flaps you will be able to lift off the runway and climb.
But back to the one-engine scenario. My gut level feeling is that any 747, if carrying no passengers or cargo and only, say, 30,000 lbs of fuel and down low (5,000 ft?) and in a clean configuration could eke out a little climb. Can't prove it though.
But start adding weight or drag and the answer would change. The first model to not be able to do it with weight and drag being added would be the 747-100. The first -100s I flew had a max takeoff weight of 735,000 lbs as I remember. The next model to fail would be the -200. 825,000 lbs was the highest max takeoff for the -200s I flew. The highest max takeoff that I have for a -400 on the DOS weight & balance software I wrote was 910,000 lbs.
The biggest difference performance-wise is due to engine size. The more power the one engine has, the more it can keep aloft.
And there are other variables. What was the nature of the failures of the three engines? Are they windmilling or seized? Even whether the remaining engine is an inboard engine (2 or 3) or an outboard engine (1 or 4) is a factor. When it comes to performance, even how many insects are splattered against the leading engine of the wing can make a difference.