How can I make 1+$MachineEpsilon not look like 1?

I think that the input field that is used to display the current value of x has its own formatting rules and altering them seems difficult. It does not follow PrintPrecision (as in SetOptions[$FrontEndSession, PrintPrecision -> 17], which works for output in an output cell).

So a workaround is to specify the Precision of the displayed x, such that the normal formatting rule shows enough digits. SetPrecision[x, 17] would be sufficient.

Manipulate[
 Row@{TraditionalForm@HoldForm@Defer@(1/(x - 1)), "\[Equal]", 
   TraditionalForm@(1/(x - 1))},
 {x, 1 + $MachineEpsilon, 2, 
   Manipulator[
     Dynamic[SetPrecision[x, 17], (x = #) &], {1 + $MachineEpsilon, 2},
      Appearance -> "Labeled"] &}]

See this answer for some further insight into how the Manipulator is formatted.