Manipulation of initial value of recurrence table

Try this. The problem is that you initial setting takes toooo long to evaluate. There is a 5 seconds limit build in.

But may be you should also limit n to something lower than 1000 like 25 or so? (Updated it to use N@t in code below, instead of just t to make it load faster initially)

Manipulate[
 ListPlot[f[t]],
 {t, -3, 3},
 SynchronousInitialization -> False,
 TrackedSymbols :> {t},
 Initialization :> (
   f[t_] := 
    RecurrenceTable[{a[n + 1] == a[n] - 1/a[n], a[0] == N@t}, 
     a, {n, 50}]
   )
 ]

So before you had the default True and that is why it aborted

Somehow the recursion poses a problem, but the iterative counterpart is rather straightforward. You can do

f[u_] := u - 1/u;
Manipulate[
 a = {N@t};
 Do[AppendTo[a, f[Last[a]]], 1000];
 ListPlot[a, PlotRange -> {-50, 50}],
 {t, -3, 3}]