One of the predefined security access masks is STANDARD_RIGHTS_REQUIRED. You see it used in defining the _ALL_ACCESS masks for various objects. Here are just a few examples: 

#define PROCESS_ALL_ACCESS        (STANDARD_RIGHTS_REQUIRED | SYNCHRONIZE | \
                                   0xFFF)
#define EVENT_ALL_ACCESS (STANDARD_RIGHTS_REQUIRED|SYNCHRONIZE|0x3)
#define FILE_ALL_ACCESS (STANDARD_RIGHTS_REQUIRED | SYNCHRONIZE | 0x1FF)

The STANDARD_RIGHTS_REQUIRED mask is meant to be used when defining access masks for object types. I'm guessing it's called STANDARD_RIGHTS_REQUIRED because it's the set of access masks that all securable objects must support. Look at the documentation or just at the definition: 

#define DELETE                           (0x00010000L)
#define READ_CONTROL                     (0x00020000L)
#define WRITE_DAC                        (0x00040000L)
#define WRITE_OWNER                      (0x00080000L)

#define STANDARD_RIGHTS_REQUIRED         (0x000F0000L)

Notice that STANDARD_RIGHTS_REQUIRED is just an abbreviation for the union of the four access bits DELETE | READ_CONTROL | WRITE_DAC | WRITE_OWNER.

Now that you see what it's for, you can also see what it's not for: You're not expected to pass it as the mask of requested access bits when you attempt to open an object. In other words, the following is wrong: 

// wrong!
HANDLE hProcess =
    OpenProcess(dwProcessId, FALSE,
                STANDARD_RIGHTS_REQUIRED | PROCESS_QUERY_INFORMATION);

The person writing this code probably thought, "Well, I just want to be able to query information, so I need to pass PROCESS_QUERY_INFORMATION. There's this other thing here called STANDARD_RIGHTS_REQUIRED; since it's required, I'll pass that too."

The "required"ness of STANDARD_RIGHTS_REQUIRED doesn't apply to you, the program opening the object. It applies to the person who is designing the object.

Your attempt to be a "good security citizen" and ask only for the access you need (namely, PROCESS_QUERY_INFORMATION) has backfired due to the addition of STANDARD_RIGHTS_REQUIRED. If you ask for STANDARD_RIGHTS_REQUIRED, you are asking for everything.

Why is that? Notice that STANDARD_RIGHTS_REQUIRED includes WRITE_DAC. If you have WRITE_DAC permission, that means that you have permission to change the security descriptor on the object, at which point you totally 0wnz0r it. You want PROCESS_VM_WRITE access but the security descriptor doesn't let you? No problem. Just set a new security descriptor that grants you PROCESS_ALL_ACCESS to the process object. Tada! You now have all the access in the world.

Moral of the story: Don't ask for STANDARD_RIGHTS_REQUIRED, because only somebody with full control will be able to get it. Ask for what you actually want.