I think this is the browser modifying the DOM due to a weird character you have in your body tag. When I view source in Firefox, I’m seeing the “character not recognized” box next to the body tag. This is also visible in Firefox inspector.
Modern browsers automatically fix issues in your HTML for you when rendering the page. For example, if you have a <table> without a <tbody>, the browser will add the <tbody> section for you, even though it isn’t in your markup. What I think is happening here is that the browser is seeing a tag that is <bodyX> where X is a bad unicode character, and it’s not recognizing it as a body tag, so it wraps the remainder of the content in a proper body tag, causing the duplication. This explains why it is not visible in view source (because view source is the literal source, not the modified source, which is visible in the inspector) and wget would produce the same behavior.
Try going into your theme’s head file and deleting the body tag, then re-typing it, to remove any strange additional characters.
If that doesn’t work, it is possible that there is something on the server that is incorrectly modifying the content of the body tag on the way out.