Above are two pictures of the same caged Kookaburra at Thorp Perrow Arboretum, photographed with the same camera and lens, from the same position, though it did turn its head between shots and the crop is slightly different. The significant difference is that the first is focussed on the mesh of the cage itself, whereas the second is focussed on the bird, causing the mesh in the close foreground to blur so much that it vanishes almost entirely – though in this instance you can just about make it out as very broad, faint bands. If you weren't looking for them you probably wouldn't notice they were there.
This is a prime demonstration that all is not lost when the fancy animal you want to photograph is behind mesh. It's important that you keep the camera aperture as wide open as possible, and that the distance between the mesh and the animal is as large as possible. Obviously if you've got a Lemur clinging to the mesh itself, making faces at you, then the effect cannot be achieved.
Unfortunately this technique is unlikely to be workable with a compact camera due to the much smaller lens and sensor – you pretty much need an SLR. You're also likely to have to switch to manual focus to stop your camera automatically focussing on the mesh, and manual focus is a facility that most compacts don't offer.