Adding counters to individual classes was discussed in some of the answers. However, it requires to pick the classes to have counted and modify them in one way or the other. The assumption in the following is, you are adding such counters to find bugs where more objects of certain classes are kept alive than expected.
To shortly recap some things mentioned already: For real memory leaks, certainly there is valgrind:memcheck and the leak sanitizers. However, for other scenarios without real leaks they do not help (uncleared vectors, map entries with keys never accessed, cycles of shared_ptrs, ...).
But, since this was not mentioned: In the valgrind tool suite there is also massif, which can provide you with the information about all pieces of allocated memory and where they were allocated. However, let's assume that valgrind:massif is also not an option for you, and you truly want instance counts.
For the purpose of occasional bug hunting - if you are open for some hackish solution and if the above options don't work - you might consider the following: Nowadays, many objects on the heap are effectively held by smart pointers. This could be the smart pointer classes from the standard library, or the smart pointer classes of the respective helper libraries you use. The trick is then the following (picking the shared_ptr as an example): You can get instance counters for many classes at once by patching the shared_ptr implementation, namely by adding instance counts to the shared_ptr class. Then, for some class Foo, the counter belonging to shared_ptr<Foo> will give you an indication of the number of instances of class Foo.
Certainly, it is not quite as accurate as adding the counters to the respective classes directly (instances referenced only by raw pointers are not counted), but possibly it is accurate enough for your case. And, certainly, this is not about changing the smart pointer classes permanently - only during the bug hunting. At least, the smart pointer implementations are not too complex, so patching them is simple.