More easily and quickly accomplished in Illustrator --
How to add multiple strokes aligned 'outside' to a type layer in Illustrator CS6?
Photoshop's Layer Style stroke always has rounded joints, unless the stroke is aligned to the inside of the layer. There's no way to change the round joints otherwise. So if rounded joints are a problem, you can't use layer styles unless the stroke is aligned to the inside of the layer. But, if aligned to the inside, strokes will cover the original image. And as strokes increase in size less and less of the actual image will be visible.
Photoshop's vector layers don't have an option for more than a single stroke. However, it would be possible to duplicate vector layers, stack them, and apply a different weight stroke to each duplicate layer, increasing in size, to accomplish something similar in Photoshop. i.e. vector layer 1 has an orange stroke of 3pt, vector layer 2 has a white stroke of 5pt, vector layer 3 has a red stroke of 10pt - as shown below.