If we have a number $x$ at a certain step, then at the next step we have $x+1$ or $-\frac 1x$. If we start with the number $1$, which of the following cannot be got after a finite number of steps?

$ 
\textbf{(A)}\ -2
\qquad\textbf{(B)}\ \dfrac 12
\qquad\textbf{(C)}\ \dfrac 53
\qquad\textbf{(D)}\ 7
\qquad\textbf{(E)}\ \text{None of above}
$
To determine which number cannot be obtained after a finite number of steps starting from \( x = 1 \) and using the operations \( x + 1 \) or \( -\frac{1}{x} \), we will analyze each option step-by-step.

1. **Option (A): \(-2\)**
   - Start with \( x = 1 \).
   - Apply \( x + 1 \): \( 1 \mapsto 2 \).
   - Apply \( -\frac{1}{x} \): \( 2 \mapsto -\frac{1}{2} \).
   - Apply \( x + 1 \): \( -\frac{1}{2} \mapsto \frac{1}{2} \).
   - Apply \( -\frac{1}{x} \): \( \frac{1}{2} \mapsto -2 \).

   Therefore, \(-2\) can be obtained.

2. **Option (B): \(\frac{1}{2}\)**
   - Start with \( x = 1 \).
   - Apply \( x + 1 \): \( 1 \mapsto 2 \).
   - Apply \( -\frac{1}{x} \): \( 2 \mapsto -\frac{1}{2} \).
   - Apply \( x + 1 \): \( -\frac{1}{2} \mapsto \frac{1}{2} \).

   Therefore, \(\frac{1}{2}\) can be obtained.

3. **Option (C): \(\frac{5}{3}\)**
   - Start with \( x = 1 \).
   - Apply \( x + 1 \): \( 1 \mapsto 2 \).
   - Apply \( x + 1 \): \( 2 \mapsto 3 \).
   - Apply \( -\frac{1}{x} \): \( 3 \mapsto -\frac{1}{3} \).
   - Apply \( x + 1 \): \( -\frac{1}{3} \mapsto \frac{2}{3} \).
   - Apply \( x + 1 \): \( \frac{2}{3} \mapsto \frac{5}{3} \).

   Therefore, \(\frac{5}{3}\) can be obtained.

4. **Option (D): 7**
   - Start with \( x = 1 \).
   - Apply \( x + 1 \): \( 1 \mapsto 2 \).
   - Apply \( x + 1 \): \( 2 \mapsto 3 \).
   - Apply \( x + 1 \): \( 3 \mapsto 4 \).
   - Apply \( x + 1 \): \( 4 \mapsto 5 \).
   - Apply \( x + 1 \): \( 5 \mapsto 6 \).
   - Apply \( x + 1 \): \( 6 \mapsto 7 \).

   Therefore, 7 can be obtained.

Since all the given options (A, B, C, and D) can be obtained after a finite number of steps, the correct answer is:

The final answer is \(\boxed{\text{None of above}}\)