As far as what it may ask and theorize about there are no limitations. As far as what it can answer, it is limited to human reason. There is no such thing as *the* scientific method, and there is no such thing as pure objectivity on the part of human observers. Furthermore, there is no pure definition of science, but only a general understanding that it consists of "the observation, identification, description, experimental investigation, and theoretical explanation of phenomena."
There is more than one theory that can adequately explain the objective universe, just as there is more than one way of doing science. The selection of theories itself is not based upon empirical principles. Even the notion that reality favors simple theories over complex ones is a philosophical principle.
What . . . is the use of making a claim you can't test?
You tell me. You make the claim that "science can only observe natural phenomena." Now test your claim. If you cannot test it then it is, as you say, scientifically "useless." Even if a claim is testable, it can only be tested within limits.
Put up or shut up.
The birth of western science may be attributed to religious assumptions regarding intelligent design. What does Darwinian evolution's disposal of any intelligent agent have to offer science? Is there something science cannot accomplish by discarding the notion of intelligent design? Also, please explain what harm has come to science by assuming an intelligent designer is behind all that science can observe and do. Put up or shut up.