When shooting through fences, a long lens (around 300mm) and a large aperture help to blur out the fence. For years I used a 300 f/4 which in many situations is ideal, but in some cases crops out part of the animal because I could not zoom out. I recently sold it along with an old style 80-200 2.8 because this version (the black Canon one) did not take extenders. I replaced it with a used Sigma 100-300 f/4 which was great until the autofocus broke and Sigma said it is too old to fix. Having learned my lesson about Sigma, I just replaced it with a 70-200 2.8 (the cheaper non-IS) which I can use with a 1.4x extender I already have. I looked at the 100-400, but I just did not want to be limited to 5.6 maximum aperture. I wish Canon made a 100-300 or 70-300 constant f/4 like the discontinued Sigma, that would be the perfect zoo lens.
I shoot only full frame which also helps blur out the fence. For technical reasons I personally do not understand, full frame sensors have a shallower depth of field than crop sensors.