You ask an interesting question, but unfortunately the spread of pathogens doesn't follow a precise course like the one you outline.
Different pathogens spread in different ways, and you also have to take into account the relative "virulence" of the pathogen, as well. Virulence is a word used to describe a bacterium's ability to infect a host. Some viruses and bacteria are considered very "weak." In other words, almost as soon as your immune system attacks them, they die off. Other pathogens are stronger (more virulent) and are better able to fend off immune system attacks.
In general, pathogens can be spread through direct contact, through the air, through water and through other organisms (such as when you get malaria from a mosquito bite). The method of transmission is unique to each type of pathogen.
In the scenario you describe, let's say someone deposits a pathogen on a vending machine button. Let's say it's a hardy pathogen that can reside on the vending machine button for several days before it dies off.
Each time a person touches the sample containing cells with the pathogen, some of those cells stick to the person's finger. Eventually, all of the pathogenic cells either will have stuck to someone's finger or will have died naturally due to exposure on the metal.
If you are the 100th person to contact a live pathogen and pick up even one cell, theoretically you could contract an illness from the pathogen. But it depends a LOT on how virulent the pathogen is, how healthy the person is, and so on.
Your example illustrates why hand-washing is so important. Your best defense against transferring a pathogen from a vending machine button to your nose, for instance, is by washing your hands frequently with good old soap and water.
Wishing you well!
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