Learn Russian: Этот батон совсем свежий.

Not quite. Батон specifically refers to a long, oval-shaped loaf of white bread, similar to a French baguette but usually wider and softer. For a standard rectangular or round loaf of bread (especially black or rye bread), Russians usually use the word буханка, or simply just say хлеб.

I know совсем often appears in negative sentences like совсем не (not at all). What does it mean here without the не?

When used in a positive sentence like this one, совсем means "completely," "entirely," or "absolutely." So совсем свежий emphasizes that the loaf is entirely fresh, perhaps right out of the oven.

Why does the adjective свежий end in -ий instead of the standard masculine ending -ый?

This is a great place to remember the 7-letter spelling rule! In Russian, you can never write the letter ы after the letters к, г, х, ш, щ, ж, or ч. Because the stem of the word for fresh ends in ж (свеж-), the ending must be spelled with an и, resulting in свежий.

Could I use the short-form adjective свеж here instead of the full-form свежий?