One common question: how complex is some aspect of the human body? In addition to directly evaluating the available evidence for that aspect, one fruitful tactic in making this kind of prediction is to analyze past predictions about similar phenomena and assume that the outcome will be similar. This is called reference class forecasting, and is often referred to on this site as "taking the outside view." 

First, how do we define complexity? Loosely, I will consider a more complex situation to be one with more components, either in total number or type, which allows for more degrees of freedom in the system considered. Using this loose definition for now, how do our predictions about human complexity tend to fare? 

Point: Predictions about concrete things have tended to overestimate our complexity

Once we know about their theoretical existence of phenomenon but before they are systematically measured, our predictions about measurable traits of the human body tend to err on the side of being more complex (i.e., more extensive or variable) than reality. 

1) Although scholars throughout history have tended to think that human brains must be vastly differently from those of other animals, on the molecular and cellular level there have turned out to be few differences. As Eric Kandel relates in his autobiography (p. 236), "because human mental processes have long been thought to be unique, some early students of the brain expected to find many new classes of proteins lurking in our gray matter. Instead, science has found surprisingly few proteins that are truly unique to the human brain and no signaling systems that are unique to it." 

2) There turned out to be fewer protein-coding genes in human body than most people expected. We have data on this by way of an informal betting market in the early 2000's, described here ($) and here (OA). The predictions ranged from 26,000 - 150,000, and that lower bound prediction won, even though it probably wasn't low enough! As of 2008, the predicted number by Ensembl was in the 23,000s. (As an aside, humans don't have the largest genome in terms of number of nucleotides either, by far. That title currently belongs to the canopy plant, pictured below (thanks to kodamatic for the photo, and to Pellicer et al. for the sequencing effort).)

3) Intro neuro texts (including one co-written by the aforementioned Kandel) claim that there are 10-fold (or more) more glia cells than neurons in the human brain. Since glia play crucial support roles and can even propagate info signals, this is not a trivial claim and would vastly increase the processing power of the brain. But when it has actually been measured, the ratio of glial to neural cells is actually around one to one in most species, including humans (see here and here). 

Counterpoint: Categories we use to explain the function of our bodies have tended to be more arbitrary than we recognize

1) One active area of research is in determining whether the distinguishing characteristics between what we consider cell "types" are more quantitative or qualitative (i.e., degree rather than form). Consider, for example, the continuum between the "classical" m1 and "alternative" m2 macrophages, which contributes whether those immune cells will be pro- or anti-tumor. Or consider the gradient of pluripotency in stem cells. If cell types are on a spectrum, depending upon the sort of transcripts or proteins they contain at any given moment, that suggests that they may be able to have more different sorts of interactions at different points in time.

2) Although we found fewer human genes than most geneticists expected, components of genes (exons) have been found to be able to combine in many ways, a phenomenon called alternative splicing. One article (here) found that of genes with multiple exons, more than 90% are alternatively spliced. Specifically, these researchers found ~67,000 alternatively spliced transcripts from ~20,000 genes. Since these alternatively spliced genes have different nucleic acid sequences, they could (and probably do) have quite different functions. 

3) The chromatin state of a given portion of the genome, i.e. where it falls on the spectrum of euchromatic vs heterochromatic, seems to have the ability to explain a large percentage of a variance in whether or not that gene is expressed. For example, one study (here) shows a strikingly high correlation between the ability of one transcription factor to bind to DNA and the chromatin state of that region of DNA (check figure 3). The fact that these chromatin states can be transmitted between generations via germ cells is also a fascinating finding that has implications which increase the complexity of human biology as compared to the "static DNA" model. 

Synthesis: When to expect more or less complexity

The above is far from systematic, but I think it portrays the trends. The known unknowns have tended to end up lower in complexity than we've predicted. But unknown unknowns continue to blindside us, unabated, adding to the total complexity of the human body. 

Why do we tend to over-estimate the complexity of known unknowns in the human body? People who study biological processes want to find more "degrees of freedom" in their systems, so that the phenomenon they're studying can have more explanatory power. The standard reason for this is that they want their results to have an impact in preventing or curing diseases, while the cynical ("Hansonian") reason is that they want to attract more status and funding. The real answer is probably a mix of both, but either way, the result is that we tend to over-estimate the complexity of the known unknowns. 

Why does it take so long to recognize the vast number of unknown unknowns? I think the best explanation for this is the standard, "Kuhnian" one, that shifting a paradigm is difficult. Adding an entirely new facet to any established scientific discipline requires slow-moving institutional support, and human biology is no exception. Look, for example, at the history of neurogenesis. Another explanation is technological, that we just don't have the capacity to observe certain things until we reach a given level of engineering success. We could not have known about histone-based epigenetics until we had the capacity to visualize cells at the level of electron microscopy (see pdf).  

The next time someone uses an argument like "the human body is so complex," try to notice whether they are referring to a prediction about the way that the human body and biology work in general, or one particular aspect of the human body. If they're referring to the general issue, at scales from the atomic to the molecular to the tissue level, they're right: there's loads we don't understand and probably lots of important stuff we don't even know about. But if they're referring to a particular as-of-yet unmeasurable aspect of the human body, past history suggests that that particular phenomenon is likely to be less complex than you might guess. 

References

Kandel, E. In Search of Memory: The Emergence of a New Science of Mind. amazon. 

Pennisi, A. 2003 Low Number Wins the GeneSweep Pool. abstract.   

Human Genome Information Project. 2008 How Many Genes Are in the Human Genome?. link. 

Pellicer J. 2010 The largest eukaryotic genome of them all? abstract. doi: 10.1111/j.1095-8339.2010.01072.x

Kandel E, et al. Principles of Neural Science. amazon. 

Azevedo FA, et al. 2009 Equal numbers of neuronal and nonneuronal cells make the human brain an isometrically scaled-up primate brain. pubmed. 

Ma J, et al. 2010 The M1 form of tumor-associated macrophages in non-small cell lung cancer is positively associated with survival time. doi:10.1186/1471-2407-10-112

Hough SR, Laslett AL, Grimmond SB, Kolle G, Pera MF (2009) A Continuum of Cell States Spans Pluripotency and Lineage Commitment in Human Embryonic Stem Cells. PLoS ONE 4(11): e7708. doi:10.1371/journal.pone.0007708

Toung JM. 2011 RNA-sequence analysis of human B-cells. abstract. doi:10.1101/gr.116335.110

John S, et al. 2011 Chromatin accessibility pre-determines glucocorticoid receptor binding patterns. doi:10.1038/ng.759.

Olins DE and Olins AL. 2003 Chromatin history: our view from the bridge, pdf. 

Wheeler A. A Brief History and Timeline: Adult mammalian neurogenesis. link.