I've read through about 100 questions/answers about the pull chain on a ceiling fan that turns on the fan (one pull), then controls speed by pulling the chain once for each speed reduction (3 speeds).

My chain isn't broken. It just doesn't do anything when I pull it.

I've read one response that I think will get me started:
1) turn breaker and the switch off
2) Remove the (bottom cap, or) light fixture if there is one (there is).
3) You'll see the switch inside attached with a thumb nut on the outside.
[? Is this the switch that holds the chain for the fan?]

4) Remove the (thumb) nut, push the switch inside and it should be exposed enough to remove the wire nuts attaching it.
[? Will it be necessary to replace the pull switch?]

5) I understand the pull switch replacement is a common item in the ceiling fan isle at most home stores.
[? Should I try to locate make and model information on the ceiling fan?]

Sounds like switch doesn't click? If not, its probably not switching. Try pushing chain into switch and then pull, if it goes on, weak spring, that should be High. Leave it there, and maybe install Fan control. I have heard several switch configurations, and direct replacement sounds difficult in some cases. Currently since no one has provided the different switch configurations, I have been cannibalizing old removed working fans and getting the switch connections. You may get lucky with replacement switch.
Before removing old switch. Write down L,1,2,3 and the colors that are attached to them.
Good Luck

I'm a little confused by your post. There are many different types of pullchains. Make/model information of your fan, or a picture, would help a great deal.

There are some older Levolier pullchains that don't click.

Ceiling fan pull chain replacement and repair - Ceiling Fans N More