I like Jay's answer, so I think I'll quote from it:
Some classification systems are logically exhaustive. Like if I say that I will divide all numbers into "those divisible by two" and "those not divisible by two", it's hard to see how you could have a number that doesn't fit into either category. Either it divides by two or it doesn't.
But many, probably most, classifications of living things are not of that sort.
This little piece from Jay's answer captures the central idea one must explore with species classification. We can often create classifications that are logically exclusive, and complete, but sometimes it just doesn't work out that way.
Something worth noting is the definition of a "species." Or rather, the lack thereof. There really isn't a good definition for species in science. It's something which scientists have found useful, but haven't found a really good way of defining. It's a bit odd, really. We're taught in science class that "species" are a real thing, but in reality they are part of a taxonomy to classify nature into nice neat buckets. What you are looking for are the corner cases that start to challenge the idea that an organism can be put into such a neat taxonomy.
In many cases the defining lines of species are very clear. In the case of sexually reproducing organisms, the ability to mate and produce a viable ofspring is a very effective delineation. Or is it? One interesting corner case is that of the Cicada. There are several species of cicada which are actually capable of interbreeding. However, there's a catch. These cicada emerge on strange prime numbered years. One set of species emerge on 17 year cycles, and the other emerges on 13 year cycles. Due to this, the organisms in these two species only have an opportunity to interbreed once every 221 years! Scientists put these in different "species" because of this, but we have to recognize that there's nothing physical preventing them from interbreeding -- its just temporal.
Once we leave the world of sexually reproducing species, the world gets even fuzzier. It is well known that bacteria trade genetic material between species. Many species have developed immunity to antibiotics by trading genetic material with other bacteria. One may argue this is as close to "breeding" as one can get with asexual reproduction, and yet we draw lines between the species.
However, we can always put the organisms into bigger buckets. While "species" may be the classification that gets the most attention, the other classifications, Life, domain, kingdom, phylum, class order, family, genus species, handle the corner cases. One might question whether M. septendecim and M. tredecim are different species (they only differ in their 13 or 17 year reproductive cycle), but both are clearly part of Magicicada, a genus which easily captures both of them.
As a general rule, the higher one goes up on our taxonomy tree (towards kingdoms and domains), the harder it is not to fit in a bucket. That's intentional... it's why that taxonomy has seen such great success. It's really hard to develop a life form that doesn't fit into a domain (Bacteia -- with their particular cell membranes, Eukaryotes -- with their bacteria like membranes but lack peptidoglycan in them, and Archaeae -- which are neither), but divisions smaller than domain offer more options. When one considers the "interesting corner cases" there's no particular reason another kingdom couldn't form, we just haven't seen one.
Talking to something mentioned by nzaman, we see few things breaking the mold of these lower level classifications because it's hard on the authors. It's hard to invent a brand new way of approaching life which, literally speaking, has not been thought of in the past few billion years. It's easier to stick to what we know. Also, if we stick to what's similar, it's easier to draw on existing emotions regarding those creatures.
However, if you're really interested in pushing the bounds, the only limit is the scientists' willingness to invent a new taxonomical category for you. If you create something that is unique enough, we'll change our tree of life to fit. It's not cast in stone. Then again, perhaps that's what you're looking for?