Does the reciprocal of a probability represent anything? I was wondering if the reciprocal of P(X = 1) represents anything in particular?
 A: Yes, it provides a 1-in-$n$ scale for probabilities. For example, the reciprocal of .01 is 100, so an event with  probability .01 has a 1 in 100 chance of happening. This is a useful way to represent small probabilities, such as .0023, which is about 1 in 435.
A: In statistical mechanics, a system has a large number of microstates, and it is a fundamental principle that these are all assumed to be equally likely. The reciprocal of the probability of a particular microstate is therefore the number of possible microstates, and this has a name in physics; it is (confusingly) called the thermodynamic probability.
The log of the thermodynamic probability is the entropy of the system, up to a constant. 
A: $\frac 1p$ does not mean anything in general (but for a particular meaning
for a specific random variable see the answer by Alex R.). However,
the 
logarithm of $\frac 1p$ to base 2, viz., $\log_2 \frac 1p = -\log_2 p$ is the amount of information (measured in bits) that you receive when you are told that
the event (of probability $p$) has occurred. If the event has probability
$\frac 12$, then you receive one bit of information when you are told
that it has occurred. In a different answer, Kodiologist has suggested that if $N$ is chosen as $\left\lfloor\frac 1p\right\rfloor$ or
$\left\lceil\frac 1p\right\rceil$, then one can say that 
$$\textrm{an event of probability } p ~\textrm{has approximately } 1 ~
\textrm{chance in } N ~
\textrm{of occurring}$$
So, since $2^{20} \approx 10^{6}$, the occurrence of
an event that has $1$ chance in a
million of occurring conveys only 20 bits or so of information
to you, far less
than is needed to transmit "Cubs win!" in ASCII !   :-) 
A: In the case of a geometric distribution, the reciprocal $1/p$ represents the expected number of throws you need to make to see one success. For example if a coin has probability $0.2$ of landing on heads, then you'd need to throw it around 5 times to see one head.
A: What are sometimes called European odds or decimal odds if fair are the reciprocal of the  probability of winning, which might be a Bernoulli random variable $P(X=1)$.
For example if the quoted odds are "1.25" and you bet $8$ then you get $8 \times 1.25=10$ back if you win (including your original stake, so a gain of $2$) and nothing back if you lose.  This would be a fair bet if the probability of winning was $\dfrac{8}{10}=0.8$, which has a reciprocal of $\dfrac{1}{0.8}=1.25$.
Similarly if the quoted odds are "5.00" and you bet $8$ then you get $8 \times 5=40$ back if you win (including your original stake, so a gain of $32$) and nothing back if you lose.  This would be a fair bet if the probability of winning was $\dfrac{8}{40}=0.2$, which has a reciprocal of $\dfrac{1}{0.2}=5.00$. 
A: In the context of survey design, the inverse of the probability of being included in the sample is called sampling weight.
For example, in a representative sample of some population, a respondent with the weight of 100 has 1/100 chance to be included in the sample, in other words, this respondent represents 100 similar people in the population.
