I'm trying a DMing style described by the Animated DM on Youtube in which one maintains a number of adventure options at all times so the PC's may go where they wish. My players want some dungeon crawls. How do I avoid having to build multiple dungeons, often?

Well there are a couple of ways to handle this. By far the easiest, is to find some sort of online repository of small dungeon maps that you can borrow from. I suggest a place like Axebane's Maps as their maps tend to be nice and easy to use with side sections for notes as needed. Otherwise, you can develop a handful of dungeons on your own and just slot one in occasionally as needed.

You can also just make 1 dungeon at a time, and whichever dungeon in the open world they go to, the map just happens to be the one you prepared. They will never know.

And if they leave the dungeon without collecting all the loot or visiting all the floors, you can just use them in the next dungeon.

There are several ways

First, be sure to have rough notes listing descriptive elements for each dungeon and have an idea of some of the monsters that might be found there. e.g. Wyrmling lair, kobold warrens, dark dwarf outpost, abandoned temple (with undead),  old mine (with cave creatures - trapper, piercer, umber hulk?, possible undead). Have an idea of whether they are constructed or natural stone. You do not need detailed maps or encounters laid out.

During play

1) Have one dungeon map. When the players choose which dungeon to go to, use that map with appropriate creatures and terrain descriptions. The players don't know that there is only one so far. You would only run into an issue if the players clear more than one dungeon in a session.

2) Have the creatures you expect to encounter depending on player choice bookmarked. In addition, jot down several possible encounters for the creature types. This mostly consists of how many creatures and possible reinforcements or other balance considerations.

Combine 1 and 2 during play to create whichever dungeon the players decide to investigate as you go along. Incorporate descriptions and possible traps based on your noted concepts for the dungeon. (e.g. there won't usually be manufactured traps in a cave system though pitfalls are possible). Refer to your notes as you go along.

The important thing is to try to imagine what the players are seeing. What are you saying and presenting. Does it fit together logically, make sense internally?

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Once you get quite experienced with this all you will actually need is a general outline and map of where locations can be found, their general character and what narrative plot elements will be present there. You can seemlessly make up everything else as you go along, as long as you are familiar with the creatures you expect to find in that locale and can judge how much of a challenge they might be.

However, it is very important to refer to your "notes" even when you are creating the dungeon as you go along since again from a player perspective, most players want to feel like they are encountering a world that is fully created with whatever is on the other side of the next door being well defined with the DM only arbitrating how the two interact.

The DM has two roles - world creator and arbitrator - most of the time this is done sequentially but it works fine doing it at the same time as long as the DM remains impartial.

However, by making the story line up as you go along, you can have a lot of interesting events happen that might not have otherwise.

Here is one example - the party encounters a gang of orcs in an underground lair, they may win. However, if the battle turns out to be more difficult than expected perhaps due to some bad luck and bad rolls. Suddenly, one of the orcs in the back row starts attacking his allies causing confusion, some of the orcs turn to face him and in the end the party prevails. If the orc dies then the party has a mystery. If the orc survives and the party doesn't decide to kill him he can explain that his mate has been chosen for the next sacrifice and the party is his only hope to rescue her. This adds a moral dilemma in terms of whether the party should help an orc. The orc may be able to provide assistance/guidance in exchange for the help he seeks. The orc may know something about the ritual about to be performed which the characters may or may not be aware of depending on their reason for being in this dungeon at this time. If the encounter is scripted or fully written down in advance it is challenging to capture everything that could happen and it takes a lot of work. In addition, if the party wins easily, this plot line and subquest may never happen or the party may never become aware that they just killed the mate of the next orc sacrifice (so why spend the time and effort creating it?).

As long as such events are logically consistent and make sense internally from the NPC point of view then the players only see what actually happens and are left to infer the rest. It can be fun to watch the players discuss some occurrence that makes no sense to them at the time but three sessions later the NPC actions make sense when the full context becomes known.

Finally, if you are creating as you go along, feel free to insert events that have no plot or storyline relation at the current time. In one game I was running, the PCs saw a huge meteor blazing across the sky during the watch at night. I just put it in at the time to make a boring watch cycle more interesting. Everyone gets up, nothing happens. The light disappeared over the horizon. There was no plot association at the time. However, about six sessions later, there was a plot element with a crashed spaceship (this was Traveller RPG), and the players connected the crashed spaceship to the streak of light in the sky at night which coincidentally tied the plot line together very conveniently.

However, if you do add elements like this you need to record them. Note taking is critical since you need to keep your world consistent, so record what you have added to the world in a specific session including unassigned plot hooks and then consider tying future plot events to those if appropriate. It gives the world a sense of reality and mystery when there isn't an immediate and obvious explanation for everything that happens.

Thanks, all! To you especially, David. What a post.

I have three or four "dungeons" in my notebook and I pick the one I am going to use based on the feedback I get from the players.  Like some other folks said, the party doesn't know you picked this dungeon.

To get them to these dungeons I plant hints that I hope will draw them in such as a portion of a map they obtain, or an NPC that hints where treasure may be found, or the party responding to an ad or rumor they heard at the tavern.

After each session I want to hear what the players are thinking about and if I don't have something like that on hand, I try to adapt what I have to that flavor.  In a real bind I guess I could make up a new dungeon just for that theme they want.

I also try and have the countryside fleshed out with towns, cities, small encampments of druids, orc strongholds, whatever.  I try and have a major backstory running that crosses the party from time to time to see if they are intrigued by the political maneuverings of the major cultures in the area.  That is something of another dungeon with the other smaller dungeons inside.

Good Luck!