There are some trick solutions
For example, in Spanish, 4 and 6 are cuatro and seis respectively, so starting at either one of those will lead to an infinite cycle of 4-6-4-6-4... Thus, I'll begin with the assumption we are just using English.
With that in mind
Negative number and decimals both have a positive, whole number of letters, so after one iteration, every number is a positive, whole integer. Every number below ten has, at most, 5 letters within it. Thus, numbers from 20-29 will contain no more that 6+5=11 letters, below the lower bound for this range. A similar argument can be applied for every other number above 29. Every number from 10-19 has, at most, 9 letters in it. This is also below the lower bound of 10. With this, it is easy to see any number above ten will quickly drop below ten.
So, for the final step
One, two, six and ten lead to three.
Three, seven, eight lead to five.
Zero, four, five and nine lead to four.
This accounts for all of the numbers. We have shown every number reaches below ten, and every number up to ten reaches four within its third step. Thus, every number will eventually reach four.
To answer your question
There is no number that does not follow this rule.