I have devised a rule that assigns a positive integer to each English word. Here is a list of words and the integers the rule assigns them.
$$ \begin{array}{l|l} \text{Word}&\text{Integer}\\\hline \text{atmosphere}&3\\ \text{day}&2\\ \text{eclipse}&7\\ \text{geology}&4\\ \text{heavy}&3\\ \text{inflation}&1\\ \text{neanderthal}&3\\ \text{parsimoniously}&4\\ \text{span}&1\\ \text{zealous}&8\\ &&\\ &&\\ \end{array} \hspace{1cm} \begin{array}{l|l} \text{Word}&\text{Integer}\\\hline \text{a}&1\\ \text{beret}&1\\ \text{cheaters}&6\\ \text{coins}&2\\ \text{dad}&5\\ \text{fad}&2\\ \text{hectares}&6\\ \text{highways}&1\\ \text{I}&5\\ \text{laziness}&2\\ \text{lid}&1\\ \text{transportation}&2\\ \end{array}\hspace{1cm} \begin{array}{l|l} \text{Word}&\text{Integer}\\\hline \text{ab}&3\\ \text{ad}&1\\ \text{bag}&1\\ \text{banana}&5\\ \text{bann}&3\\ \text{hear}&4\\ \text{her}&3\\ \text{males}&1\\ \text{prevent}&1\\ \text{snap}&1\\ \text{thorny}&1\\ &\\ \end{array} $$
The rule can be described in a single sentence.
What is my rule?
Edit: I have chosen some additional words and computed the numbers the rule assigns them. Also, find some hints below.
Hint 1:
It is unlikely you will find a word in the dictionary that is assigned a value larger than $10$. However, the rule is defined for arbitrary strings of English letters, and it is possible write down strings with arbitrarily large output. To get an output of $11$ requires a string of at least 16 characters, and the 16 character string with output $11$ that comes first alphabetically is $\text{'dzzzzzzzzzzzzzzz'}$.
Hint 2:
It might be useful to start by figuring out what the words assigned $1$ have in common.
Hint 3:
The US coins are: penny (1¢), nickel (5¢), dime (10¢), quarter (25¢), half dollar (50¢), and dollar (100¢). This is relevant.
Hint 4:
The rule assigns anagrams the same number.
Hint 5:
The rule does not depend on any of the following:
Case, font, keyboard layout, letter frequency, word meaning
Hint 6:
The longest string that the rule assigns $1$ has one hundred letters.