This is a very important question and the answer varies widely between individuals and breeds!
A generally and logical guide is to avaoid all excessive excercise or activity until the bones are mature - that is, that the bone growth plates have closed (or fused). But what is "excessive" ...and what are the "growth plates" ?
When puppies are developing in thier mother's uterus, they initially have NO bones at all, and their "bones" are actually made of cartilage. Gradually this cartilage calcifies and becomes bone, and it does so by starting at specific points and spreading outwards. Also the long bones, like the upper and lower leg bones (but also wrist, paw and foot bones) actually have growth plates which actually grow bone and thus grow the length of these bones. The length and thus also the height of a puppy is dependent on how long these plates grow bone before they stop doing so - at which point they "close" or fuse.
If these growth plates are damaged in any way, by extreme activity or an accident (trauma), they will prematurely close and stop adding to the length of that particular bone. This causes the affected bone to be shorter than the same unaffected bone on the other leg. IF the damaged growth plate is in the upper front leg (humerus) or the upper hind leg (femur) bones, then all that happens is that leg will end up shorter than the opposite leg. A slight limp may be the result.
IF the damage is in a one of the PAIRED bones of the legs, particularly the front leg (the raduis and ulna), then the result can be very debilitating indeed. This is because one of the paired bones with the damaged growth plate will be shorter than the other bone, causing an angular deformity at the wrist (carpus) and severe pain in the elbow as the shorter bone (especially if its the ulna) will crush in at the elbow joint. The more common bone affected is the ulna, and it is more common in small to smal-medium breeds (although ANY size dog can develop this problem).
Some other smaller areas of the leg bones may have a delayed, slowed change from cartialge into bone and inappropriate exercise can cause a fracture through this weak point (cartilage is far weaker than bone). Some dog breeds actually suffer a disease that stops the full change of cartilage into bone in specific areas. In particular German Shepherds are more likely to get a disease called "un-united anconeal process" and Rottweillers are more disposed to "fractured medial coronoid process". Both are in the elbow.
Excessive exercise is anything beyond what a normal puppy in its litter, with its mother, would normally do. Hmmmm.... I hear you say!!! Another "How long is a piece of string answer" !!! Well, normal play activity ... but of course you can begin lead walks to the park, and allow normal play there as well.
The BIG NO-NO is "forced exercise". Taking your puppy for an extended jog or run ... having your puppy race along while you cycle ... extreme games like throwing frisbees, balls and sticks (especially if they are leaping to catch and thus may land awkwardly). These sorts of activities should be delayed until AFTER THE BONE GROWTH PLATES CLOSE. The concussive forces coupled with the shearing forces are very dangerous for growth plates and delayed calcification areas.
Another activity to delay is Agility training. Most agility trainers are well aware of the potential for joint and bone damge in young dogs and puppies and will advise you to delay progressing from obedience to this - Please follow their sound advice!!!
In small breeds, this is around 9 months of age. Medium breeds it is 10 -11 months. Large breeds it is 12 - 14 months. For giant breeds it is 18 - 24 months.IF IN DOUBT, delay the planned extreme activities for an additional month or two!
I will add at this point, to ensure excellent nutrition of your puppy - ESPECIALLY LARGE AND GIANT BREED DOGS. I advise one of the Super Premium Large Breed Growth or Puppy diets - Hills, Advance or Royal Canin are my preferred. A second related point is to not feed too much. If you over feed, then a puppy's growth rate will be extreme and this more likely to cause developmental bone and joint problems, particularly of the cartilage to bone calcification that leads to the Un-united anconeal process and fractured coronoid process diseases.
OCD, or Osteo Chondrosis Dissecans, is a related disease where the cacification of the cartilage under the joint surface is affected. Because the calcification doses not extend to the joint cartilage (articular cartilage) the area remains as thicker cartilage, which is weaker than bone, and will fracture at the interface of the deep/thick cartilage and the underlying bone. This causes a flap of thick cartilage, into which joint fluid gets compressed with activity, getting "pumped" into the fissue by the movement of the limb, causing extreme pain.
So, hold back on excessive activity, allow the normal puppy behavioural play and lead walks. Then progress to obedience training and socialisation at the parks! Have fun and enjoy your puppy as it grows into a healthy dog!