An anti-razor is, as the name implies, the opposite of Occam's Razor, and states that we should continuously add entities until we can support our argument.
Or as stated by Walter of Chatton, a contemporary of Occam, "If three things are not enough to verify an affirmative proposition about things, a fourth must be added, and so on". So, instead of entities (assumptions) being "shaved off", which, incidentally, is where the "Razor" in Occam's Razor gets its name, we are adding entities.
These two principles are usually applied in the search for the simpler of multiple explanatory theories for some phenomenon. The introduction of assumptions such as, for example, the statement that encountering fossils of non-extant organisms similar to other living organisms proves that they are "precursor" or ancestral organisms, as opposed to simply distinct organisms that just happen to be similar, violate Occam's Razor and should instead be treated as separate hypotheses on their own. These hypotheses, if provable (though in our specific example it is scientifically impossible, as you cannot observe this transition taking place) can then be used as one of the bases for our argument, but if refuted should not be able to affect our theorem, as they should be treated separately.
Also See Russell Wright