Behind my urban apartment building is a five story high, and still growing, silver maple. It is starting to break the concrete wall it is next to. Short of spending thousands to cut it down, which would be a shame, is there a way to stop it growing? Maybe cut into the roots and lop off high big branches?

Possibly. Utility companies stunt fast growing trees with gallons (a cup at a time) of paclobutrazol. They drill down all around the root edges and pour the chemical in the holes. But what's your water supply? Any fish ponds? Laws about chemicals? We're talking GALLONS. I would much rather build the wall in an arc around the tree, if there's room to move it back. Pruning and cutting roots isn't that simple. You could discuss it with a tree surgeon.

How close is it to your building? What type of a wall is it damaging? If the wall is essential (such as a retaining wall),a new wall will likely require a footing at a depth that will likely damage a significant number of roots Walls without deep footings (several types in use) will be expensive. Sounds to me like the best approach is to take the tree down and hopefully before the wall is damaged beyond repair. I don't enjoy removing trees but sometimes it is the only cost effective solution. You can find some relatively inexpensive quick maturing replacements. Especially if it is close enough to damage your building and any adjustments to the tree (except normal pruning ) will increase the chance of it falling at some point in the future.