How does one get a picture similar to the one below with a crop sensor DSLR and using kit lens or 70-300mm lens without post-processing?
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The trick is to use a long lens (the longer, the better), then get meaningfully far away from your foreground subject matter.
The long lens will make both the moon and the foreground appear much larger than they otherwise would. The narrower field of view does that for you. Moving further away from your foreground subject will make it appear smaller in the frame. That much should be obvious and intuitive. Double your distance, and the object becomes half of its previous size using the same focal length lens. The difference between 100m and 500m is huge. The thing is, though, that you can't make much of a difference in the distance between you and the moon no matter how many steps backward you take. The difference between 380,000km and 380,000.4km might as well not be there at all. So while the moon stays the same size, your foreground objects become much, much smaller.
If you cannot get the geometry to fit for practical reasons, you can double expose the shot. It is a trick used even in the old days with film, if at night with a black sky.
First imagine your shot in your head. Then take your widest angle view shot with the exposure that is nice for that. Then zoom in one the moon and expose it nicely, and place it in the frame where you want it. If at night it is easy to overlay these simply by adding them together or applying a "max" function. With the clone tool you can adjust the position of the moon as well.
If it is not dark you need to mask it in.
Here I made a fun, surreal example with a Zeiss Jena 135mm for the moon and a Pentax 50mm for the lamp post: