Print paper or cellulose - any recyclable filament?
Filament made of Polylactic acid (PLA) is usually made of biological materials (such as corn), and can therefore be considered bio-degradable in most cases.

Whether the filament is 100% bio-degradable (and non-toxic for the surroundings) will depend on the specific formula used by each individual filament manufacturer. (Many manufacturers include various additives to achieve particular effects, such as glow-in-the-dark, metallic finish or extra strengh.)

Woodfill PLA-like filament might be of extra interest to you, not only because it typically is bio-degradable, but also because it will give you the look and feel of being bio-degradable. 

Hope that helps!

PS: there are multiple other filament types that are either recyclable, bio-degradable, or both, although PLA might be the most commonly available of them all.