The existence of Giffen behavior, behavior where consumers demand more of a good when the price increases, is controversial. Theoretically it requires that "the good in question be strongly inferior and that expenditure on that good comprise a large portion of the consumer’s budget," but actual examples have been hard to find.
This paper looks into the theory behind Giffen behavior and provides evidence that it exists (see footnote 1 for why they use the term Giffen behavior rather than Giffen good, and their warning to be sure to distinguish Giffen goods from "prestige or Veblen goods, where price signals quality and/or consumers desire the goods precisely because the price is high"):
Giffen Behavior: Theory and Evidence, by Robert T. Jensen, Nolan H. Miller, NBER WP 13243, July 2007 [open link]: I. Introduction The “Law of Demand,” which holds that as the price of a good increases, consumers’ demand for that good should decrease, is one of the bedrock principles of microeconomics. However, economists have long recognized that the axioms of consumer theory do not guarantee that demand curves must slope downward. Alfred Marshall first publicized this idea in the 1895 edition of his Principles of Economics:
As Mr. Giffen has pointed out, a rise in the price of bread makes so large a drain on the resources of the poorer labouring families and raises so much the marginal utility of money to them, that they are forced to curtail their consumption of meat and the more expensive farinaceous foods: and, bread being still the cheapest food which they can get and will take, they consume more, and not less of it. (p. 208)
Taking the introduction a bit out of order, here's a description of Giffen behavior:
The conditions under which we would expect Giffen behavior can be demonstrated by elaborating Marshall’s statement. Imagine an impoverished consumer near a subsistence level of nutrition, whose diet consists of only two foods, bread and meat. Bread offers a high level of calories at low cost, while meat is preferred because of its taste (but provides few calories per unit currency). The consumer therefore eats a lot of bread in order to get enough calories to meet his basic needs, and with whatever money he has left over, he purchases meat. Now, if the price of bread increases, he can no longer afford the original bundle of foods. And if he increases his consumption of meat, he will fall below his required caloric intake. So instead, he must increase his consumption of bread and cut back on meat.
Thus, although the price increase makes the staple relatively less attractive, the fact that the price increase makes the consumer so much poorer (in real terms) forces him to consume more bread. Translating this to the language of consumer theory, the conditions under which Giffen behavior is likely to be observed therefore include that the good in question be strongly inferior and that expenditure on that good comprise a large portion of the consumer’s budget. ...
Since Marshall’s time, a discussion of “Giffen” behavior has found its way into virtually every basic economics course despite there being no evidence that the demand for either bread or wheat was upward sloping in Britain during Marshall’s time (Stigler 1947 and Koenker 1977). The standard textbook example of a Giffen good, the case of the potato during the Irish famine of 1845-1849, has also been discredited. Not only are there no data to support the claim (Stigler, 1947), but at a more basic level it is unlikely that consumption of potatoes could have increased when the price rose during the famine, at least in the aggregate, precisely because the price increase was caused by a shortage of potatoes due to a blight that destroyed much of the crop.
The fact that there has to date been no convincing evidence of Giffen behavior stands as a minor embarrassment to economists... This lack of evidence has prompted a range of reactions among economists. Some have ... been more extreme, interpreting it as an indictment of neoclassical consumer theory. Along these lines, Boland (1977) points out that not only is the theory unable to rule out Giffen behavior, it is also unable to explain why Giffen behavior is not observed. Put another way, if the neoclassical model is correct, then under certain (albeit uncommon) conditions, Giffen behavior should exist.
If it has not been observed, it is either because the appropriate conditions have not been satisfied, the appropriate data have not been available to measure it, or our theory of the consumer is incomplete or flawed.
One of the primary challenges in searching for evidence of Giffen behavior is finding sufficient and exogenous price variation for staple food items. ...
And they claim to have done just that. Here's the abstract summarizing the findings:
Abstract This paper provides the first rigorous, empirical evidence of the existence of Giffen behavior... We begin by examining several theoretical approaches ... and show that ... Giffen behavior is closely associated with poor consumers' need to maintain subsistence consumption in the face of an increase in the price of a staple commodity. We then present evidence on the existence of Giffen behavior among extremely poor households in two provinces of China. In order to obtain an unbiased estimate of the key price elasticity, we conducted a field experiment in which we randomly subsidized households' primary dietary staple (rice in Hunan province and wheat flour in Gansu province). Using consumption data gathered before, during and after the intervention, we find strong evidence of Giffen behavior with respect to rice in Hunan province. We also find evidence for Giffen behavior in Gansu with respect to wheat; however, the evidence is less robust than for Hunan, due to the (unanticipated) failure of at least two of the theoretical conditions that appear necessary for Giffen behavior. Restricting the Gansu sample to households that meet these conditions provides stronger evidence of Giffen behavior.
 We use the term “Giffen behavior” rather than “Giffen good” to emphasize that the Giffen property is one that holds for particular consumers in a particular situation and therefore depends on, among other things, prices and wealth. Thus, it is not the good that is Giffen, but the consumers’ behavior. The Giffen phenomenon should also not be confused with prestige or Veblen goods, where price signals quality and/or consumers desire the goods precisely because the price is high. Giffen behavior is a phenomenon that arises entirely within the neoclassical framework where consumers care about price only inasmuch as prices affect their budget sets, which rules out prestige goods.
 Another argument notes that with upward sloping demand in stable equilibrium (i.e., supply is flatter than demand), the supply reduction due to the famine would actually lower the price of wheat, not raise it. Dwyer and Lindsay (1984) present a summary of the basic case against the potato version of the Giffen paradox.
 Perhaps the best evidence to date is by Battalio et al. (1991), who find evidence of upward sloping demand curves among rats given limited ‘budgets’ and the choice between root beer and a quinine solution.
 Others have argued that it is not our understanding of consumers that is flawed, but rather our understanding of markets. For example, Dougan (1982) argues that markets with upward sloping demand curves are inherently unstable, and thus unlikely to be observed, while Nachbar (1998) shows in a general equilibrium framework that observing the equilibrium price and quantity of a good move in the same direction in response to a supply shock