Why don't companies just print more of valuable items?

Do they have some kind of written or unspoken agreement saying that they will never do things like that...

Correct ...

I realize that the value would drop if they did, but for them, it would still be worth it in terms of money?

Ahh ! Don't forget, if they did that: people wouldn't buy them in the first place!

You see?

Say, today, they suddenly decided to print another 100,000 Pikachu Illustrator cards, and sell them for five bucks each.

Sure, it might work "today" - they might sell a few thousand of them ("to suckers!"), gaining the odd few thousand bucks for the company.

BUT from then on, people would say "these cards are worthless, there are no rare cards, there's no point collecting them".

So, "that day" ALL the Pikachu Illustrator cards (there are like 20 of them right?) would be worth nothing, rather than they are currently worth $100,000 each. So those 20 people would be pissed! But even worse, everyone would now say "it is pointless collecting Pokemon cards, they just print them".

If gold were as common as lead, then it wouldn't be worth much more than lead. (Sure, it's shiny, minimally reactive and conductive, so would be worth more than lead, but not much more.)

For this very reason, King Henry IV of England outlawed alchemy: more gold means less valuable gold, and therefore less wealth for the people already owning gold.

The Pokémon Company deciding not to print more of those cards is following exactly the same economic principle.

With collectibles, supply and demand are the primary factors that determine price. An item becomes a collectible when there is demand and the item is somewhat rare.

If you now have a valuable collectible and you increase the supply, not only does the value of the collectible decrease but so does the demand.

So if Pokémon were to print more of their most value cards, not only would that devalue those specific cards but it would diminish interest in all of their cards because the chances of finding those rare cards gem would be lower. That in turn would mean that people would buy fewer cards. It's just basic economics.

Relevant(?) anecdote:

On October 23, 1962, the US Postal Service issued a stamp commemorating the late UN Secretary General Dag Hammarskjöld. About 121.4 million of the stamps were printed.

It was soon noticed that an unknown number of the stamps had been printed with the background inverted, creating a rare and valuable error. One such stamp, used on the day of issue, sold in 2005 for $US3500.

On November 16, 1962, the Postal Service issued 40.3 million of the stamps printed with the background deliberately inverted. The stamps were indistinguishable from the original error, and diluted the rarity and value of the error.

See https://en.wikipedia.org/wiki/Dag_Hammarskj%C3%B6ld_invert