Can a minimum or maximum be an outlier An outlier is defined as 1.5 times the IQR added to Q3 and subtracted from Q1. My question is this, is it possible for an outlier to be a maximum or a minumum value when making a box plot?
Yes the max and min can be outliers. In a boxplot of the style that can show outliers, the 'lower fence is at Q1 - 1.5(IQR) and the upper fence is at
Q3 + 1.5(IQR). Ordinarily, fences are not plotted.

The lower 'whisker' extends downward to the the lowest observation that is still above the lower fence. The upper 'whisker' extends upward to the the highest observation that is still below the upper fence fence.

Exponential data are strongly right-skewed (tail extending toward higher values) and there is often at least one outlier on the high side. To illustrate, I use a sample of size $n = 50$ from an exponential distribution with mean $\mu = 10$ and a sample of size $n = 50$ to illustrate a boxplot.

Rounded to integers and sorted in increasing order, the observations are:

  0  0  0  0  0  1  1  1  1  1  1  2  2  2  2  2  2
  2  3  4  4  5  5  5  6  6  7  8  8 10 12 12 12 13
 13 14 14 15 16 17 18 19 19 20 22 23 25 28 33 38


The upper quartile is at 14.75 or 15, depending on the definition used. I use $Q_3 = 14.75$ and IQR $ = 12.74,$ so the upper fence is at 33.875. (There
are about 10 different rules for quartiles in common usage among various textbooks and software packages.)

The boxplot is shown below, and the upper fence is added (vertical dashed line).



The second highest observation 33 is just lower than the upper fence, so
33 is the end of the upper whisker. (It is incorrect to call the end of this whisker a 'maximum bar' because it is not at the maximum.) The minimum is at 0 and it is not an
outlier. The maximum is at 38 and it is an outlier (dot at the right).
Typically, when making a box plot, you will first identify the outliers in the data, then draw the box plot using the maximum and minimum of the data remaining after you remove the outliers. As a simple example, suppose you had the data $\{0, 4, 5, 6, 9, 13, 1000\}$, with 1000 being the obvious outlier. When you draw the box plot, you would put the maximum bar at 13, and a small circle at 1000 to mark it as an outlier (plus, you would probably put a little mark in to allow a discontinuity to let you cut out the space between the two).

The diagram below shows how you would do that.