How big is big to you? Dick Robertson, Paul Brunner, Dean Torges, John Strunk, Ted Frye (Raptor Archery, Jay Massey and many many others all shot the biggest of North American game animals with selfbows or backed bows(sinew or rawhide, etc)....Elk, black bear, grizzly, moose, whitetail....they and many others have done it all.
Go through an edition of Primitive Archer magazine and can see everything from small game to African plains game....cougars to bobcats.....coues deer to 6x6 elk...and even Yukon moose. Though the magazine is more about sharing knowledge and skills there are also a few hunting stories and hero shots.
Here is search link with eland, antelope, manitoba bear, moose, bull elk, hogs, whitetail, pronghorn antelope, cougar, javelina, wild boar, kudu, wildabeest, and more
https://www.google.com/search?q=osage+selfbow+trophies&safe=off&client=firefox-a&rls=org.mozilla:en-US:official&source=lnms&tbm=isch&sa=X&ei=KizdUpqUL6qtsATk7oCoBw&ved=0CAcQ_AUoAQ& biw=1920&bih=904