How many semicolons is too many?

It depends on how the semicolons are being used.

Semicolons are, in general, used two ways:

• to link together two clauses in a way that eliminates the breath space of a period, and
• as a way to separate items in a list where commas are also present

Used the first way, I'd say anything more than one semicolon is too many (unless you are deliberately creating an atmosphere where you want the reader to not breathe to create tension). For example, if I write this:

I wandered down to the pond out back of my house; the ducks were lined up as if waiting for me; my shotgun lay in the crook of my arm, cold, deadly; dinner tonight was going to be peppered with pellets.

The ideas I'm trying to express sort of start to lean into each other - the relationships between the pond, the ducks, the gun, and eventual dinner all bleed over (no pun intended). Some periods would help define the mood, like this:

I wandered down to the pond out back of my house; the ducks were lined up as if waiting for me. My shotgun lay in the crook of my arm, cold, deadly. Dinner tonight was going to be peppered with pellets.

A couple of periods and a couple of capitalized letters and we have a mood that is better defined and, for me at least, there is an ominousness in the second example that simply wasn't achieved in the first.

When using semicolons in a list, I'd say your limit is not with the semicolons but with the number of items in the list - generally a list can have three or so items and not get bogged down but it is possible to have longer lists that read just fine. For example, I can write:

I was so excited to finally be going on my first big adventure that I packed a week early! Not really knowing what to expect, I kinda packed everything: my big, brown, wooly slippers; six pairs of pants: blue, pink, indigo, yellow, a floral explosion, and beige chinos; six tops of various styles and colours - probably none of which go with my pants but I can always go shopping and get more; shoes, shoes, and more shoes!;...

and we can go on - the list itself providing a kind of narrative in this little example of a story.