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The fact that your plot doesn't show "too high" values isn't specific to LogPlot, has nothing to do with the PlotRange you specified and also isn't due to the values being "too high". It is the result of the initial sampling of the plot points. According to my observations so far, no Plot includes the lower and upper limit specified for the variable. For example Plot[f, {x, xmin, xmax} plots f for xmin < x < xmax and not not for xmin <= x <= xmax, meaning f is plotted for the open interval $(xmin, xmax)$ and not for $[xmin, xmax]$.

First one way to make your LogPlot look like your ListLogPlot

First let's define a function that will print out its input and evaluate to 1

The fact that your plot doesn't show "too high" values isn't specific to LogPlot, has nothing to do with the PlotRange you specified, and also isn't due to the values being "too high". It is the result of the initial sampling of the plot points. According to my observations so far, no Plot includes the lower and upper limit specified for the variable. For example Plot[f, {x, xmin, xmax} plots f for xmin < x < xmax and not for xmin <= x <= xmax, meaning f is plotted for the open interval $(xmin, xmax)$ and not for $[xmin, xmax]$.

First, one way to make your LogPlot look like your ListLogPlot

Let's first define a function that will print out its input and evaluate to 1

There is an extensive How does Plot work? Q&A that already analyses the inner workings of Plot. However, the problem that the function isn't plotted for xmin and xmax isn't part of it.

There is an extensive How does Plot work? Q&A that already analyses the inner workings of Plot. However, the problem that the function isn't plotted for xmin and xmax isn't part of it.

Generalizing the Problem

does not start at x == 0 and end at x == 1(, it starts at x == 2.040816326530612*^-8 and ends at x == 0.999999979591837). The only difference is, that here it isn't as obvious as in your 1/x case.
It is also worth pointing out, that this is not limited to the case where xmin is 0.

The problem generalized

does not start at x == 0 and end at x == 1, it starts at x == 2.040816326530612*^-8 and ends at x == 0.999999979591837. The only difference is, that here it isn't as obvious as in your 1/x case.
It is also worth pointing out, that this is not limited to the case where xmin is 0.