This is not possible. And in fact, you should really want the
Save As to appear when it does.
Save feature only works when you are editing a document that you have open. In other words, same title, same suffix, and all layers/features are compatible with that format still.
Save Asappears when you have used features in Photoshop which are not supported in the existing document's format. For example, you open a jpg and add 2 layers... that can no longer be jpg because jpgs do not support layers. So
Save isn't available and the
Save As window appears. It's not about having to "save as psd", it's about what a particular image format supports.
To circumvent this behavior is asking Photoshop to have the ability to randomly overwrite any file at any time and ignore any issues with internal compatibility with all formats.
In other words.. you spend 8 hours working on a 120 layered, CMYK, smart object file and hit
Save, saving it with a .gif suffix replacing another file. You have then essentially made the resulting .gif file useless because it contains data which the gif format does not support so it can't be opened or used anywhere. So, you've just wasted all that editing time with no way to recover the work.
I understand that taking the extra step to deal with the
Save As dialog may seem like a time waster... however, much like the airbag in your car... it's there for a very important reason. Trying to turn it off or get around it is asking for trouble whether or not you realize it.
All that being posted... you can generally open any file, do anything you want to it... then choose Flatten Image from the Layer Panel Menu and chances are
Save will then work. (Because you've reduced everything down to a single layer, flattening any vector data, shape layers, smart objects, layer styles, etc.... which is what most image formats support.