You always here how chimps are humans closest relatives, having 96% (or 98% or whatever, depending on the source)
OK fair enough, but what animal are we LEAST releated to. I guess by this I would say what animal do we have the least percent of DNA in common with.
Of things classified as “animals”, it looks like the Porifera or sponges probably split off from the rest of us the longest time ago. We are also related to fungi, plants, protozoans, and ultimately even the various groups of “bacteria”.
On the assumptions that the OP really did mean ‘animal’ (as opposed to using it as shorthand for ‘any lifeform’) and the probably univesal consensus that what used to be termed proptozoa are not animals but several groups of protists, if we add the third assumption that the answer has to be something that anyone not a specialist has ever heard of, the answer would be, any randomly chosen sponge. The entire sponge phylum is sufficiently different taxonomically and evolutionarily from the rest of the animals that it’s kind of like arguing which of two points within the city of Yakutsk is closer to Miami Beach – any selection of specializations to claim that this lor that lineage is less closely related is pretty much arbitrary.
If you want to get truly arcane, the one group even more distant than the sponges is the Mesozoans – microscopic endoparasites and commensalists on, IIRC, molluscs, that are best described as being a plucky young group of colonial microorganisms that decided to jointly play a multicellular animal summer stock, but are doing a really poor job of acting. 
It’s more like arguing whether your niece is more closely related to you than your nephew. As far as we know, all sponges had a common ancestor that wasn’t also our ancestor, and no matter how many species have evolved from that common ancestor, they’re all equally distant from us.
I thought popular opinion was that they were degenerate flatworms.
Colibri or Darwin’s Finch may have something more definite on the Mesozoa – when last I read up on phylogenesis – admittedly, several years back – the general view was that they were specialist survivors of a separate early, failed attempt at multicellularity, with the “degenerate platyhelminth” position a strongly held minority view. Good point, though – I’d forgotten the argument until you brought it up.
What about either those critters that live down in the bottom of the marianas trench sucking up stuff from the volcanic vents? Or those floaties they found down in IIRC antartica that had been in that isolated frozen lake for the last hundred million years or so?
The most advanced animal, in terms of intelligence, that we’re least related to, would be the octopus.
Widely thought to be the most intelligent invertebrate.
Any other candidates?
From what I can glean, the present consensus (although not universally accepted) is that they are at least Metazoans (which includes almost all animals except the sponges, coelenterates, and comb jellies), and seem to be closest to some groups classified among the flatworms. See here for a recent paper (pdf).
I’m not sure exactly what you are referring to. If they are multicellular animals (and not sponges), then they are more closely related to us than sponges are.
Beyond the sponges, we are progressively less closely related to the fungi; multicellular plants; the various unicellular “protist” groups; and the archaea (bacteria-like organisms). The organisms that are least closely related to us are bacteria.
Richard Dawkins’ book The Anceztor’s Tale imagines us walking back into history and meeting up with lifeforms with whom we share a common ancestor. We meet chimps and bonobos first, then gorillas then orangs and so on back to … The sponges (then plants then fungi then bacteria).
Each animal we meet has a little tale to tell of some characteristic that makes humans whAt they are. Fascinating stuff.
Best book of the century (so far) in my opinion.