Is there a term for the datum with the least (non-zero) frequency? The mode of a (finite) data set is the value with the most number of appearances. What about the opposite of the mode? That is, the value (from among the data) with the least number of appearances? Is there a term for it?
 A: The answer is contingent upon the type of data. 
Case 1 
If the data is quantized with a finite number of observations, for example during counting of nuclear decays, then although we may approximate it as a continuous variable it is in actuality  a series of Dirac $\delta$ occurring at distinct $X$-axis offsets. That is, if each event has an equal probability of occurrence and an area of $1/n$ for $n$ observations with a width of zero, then each observation is both the minimum and maximum probability occurring at that $X$-axis position. It is only when we look at the local (spatial or temporal) frequency of those events that we give rise to histograms or kernel smooths that look like continuous probability density functions. It is from the local density of events that a mode is identified, at a given smoothing. When we restrict ourselves to examining single events, while ignoring non-events, any pretense of uniqueness is lost, and in the case of such lists of non-unique one dimensional events the name for that is
List mode data. And, even when binned the data is still discrete. In that case, the name for the non-zero minimum of one event is all of the data When smoothed (e.g., kernel smooth) the data then becomes more continuous.
Case2
Let us consider data which whether quantized or not is being measured as wave functions or other continuous or very nearly continuous methods. For example, we might argue that length is best measured to the nearest Planck length or approximately $1.616229×10^{−35}$ meters since space is relativistically invariant for discrete numbers of actual Planck lengths, where as a meter stick changes length depending on the observer's frame of reference. Ignoring for the moment whether or not a true continuous form of raw data exists in nature, or whether there is a noise or "granularity" limit, there is no discounting that we can approximate physical measurements using continuous functions.  
Then we can argue that the probability of a non-zero continuous pdf when the support is at least semi-infinite can be made as small as we like over a semi-infinite span. This does not seem like it has any unique "anti-mode." Indeed, if the local probability density is so close to zero that it is vanishingly small, then it can also be so small as to be irrelevant. So, in the case of some continuous data, the concept of an "anti-mode" is not practical, and mathematically, we could say it is "at infinity." Now consider infinite support, in that case the "anti-mode" is at $\pm \infty$, which seems neither here nor there nor very useful.
There are undoubtedly other cases. However, for both Case 1 and 2, most often there will be a unique mode, but no useful non-zero opposite of a mode. Thus, if there is such a thing as an "anti-mode" there are many cases in which it is not as useful as the concept of a mode.
