What is the heaviest mechanism? I'd say there are a few clear contenders that all share similar traits. But I'd like to point out one mechanism that while not the absolute heaviest in existence is far more difficult than it gets credit for.
Most of the 'heaviness' of games comes from how mechanics fit together—either their resulting rules overhead or their emergent strategy, depending on which you qualify as being heavy. Individual mechanics are generally pretty light, unless they require multiplication or division. The heaviness of strategy games generally comes from spinning a lot of plates.
Ideally, each mechanism in board game design could be 1) performed on one turn and 2) explained with one line of text. But some mechanisms need context to reveal themselves. So, "pickup resources" and "deliver resources" combine to become something more distinct as "pickup and deliver." And mechanics that reveal themselves thru context cannot usually be explained by a single line of text. I classify these mechanics as inherently heavier.
For my money, the various multi-phase auction mechanics are probably the heaviest mechanics currently in the BGG database (but I'm not combing thru all of them or even all of the auctions to give a definite winner). They require the context of multiple players over multiple bids before they can be understood. This in turn requires a more in depth rules explanation. (They also require implicit understanding of the value of items and a certain amount of mental math and probability tracking to strategize well.)
The mechanic I want to bring to your attention is nearly as difficult: multi-turn, multi-player context is required. It requires a paragraph of rules text. It has certain unintuitive conditions that trip up new players. It's pick-and-pass drafting.
Pick-and-pass drafting (here on out, drafting) is a mechanic that has to be described in full, across an entire round and often into the following round before players have full context of what they are doing, let alone why. Just telling them to "take one card and pass the hand to the left" leaves out what you do with the card you take and whether you can swap that card for a future card. I'm blaming Spoons for making drafting so difficult to teach. (At least with American players above a certain age.) In Spoons, you keep a hand and pass a single card. There is only one hand that is yours. In drafting you can have your tableau, your hand, and the hand of cards you are passing. Players have to be taught explicitly what cards they own, what are locked in place, what gets passed, what can be discarded. And then they have to not get those categories confused. Which is difficult because it's all the same deck of cards. How many times have you played Sushi Go! only to have a player run out of cards before everyone else? For me, it's every time.
To add to the rules complexity, most drafting games have additional rules around the direction of the draft. Whether it's left-right-left or a snake draft, these rules add complexity to an already complex system. Then you get the 2-player variants, which require you to understand the original rules so you can know what to do differently. I play a lot of games 2-player on the first play, so this standard of rules writing is maddening. Give me the whole turn structure in the variant section.
Designers tend to think of drafting as a simple mechanic. Then they add it to games to increase complexity (e.g. Terraforming Mars). As someone who has designed a drafting game, how we are using the mechanic is more in line with what drafting really is: a complexity adder. In my experience, drafting is heavier than engine building. Both require knowledge of all of the cards and what they do, but drafting requires that knowledge to be given beforehand since the draft isn't open the way placing a card on the table is. This is why card glossaries are necessary. I can hand the rules to a player and tell them to look up the card in their hand, rather than having to explain out loud what the card does. Pool drafting is inherently lighter because the drafting is open on the table but also because it can be explained as a single action. (For what it's worth, "I cut; you chose" is heavier than drafting—because you are predicting other players' wants—but is often open information.)
Engine building generally doesn't generate player interaction (expect maybe when you run your engine). Drafting requires consideration around not only what you are keeping but what you are giving away. Not only do you have to understand all of the cards, but also their values at given points in the game and what your neighbor may find valuable based on what they have played thus far. We've racked up a lot of spinning plates for a single mechanism.
We often assume that everyone in the hobby has played a drafting game. I have playtested with a lot of designers who had never played a drafting game. I rarely had a playtest where I didn't have to teach "how to draft." Drafting is by far the heaviest mechanic in my game. Even if you aren't designing a drafting game, be aware of how context creates complexity. The longer something takes to explain the heavier it is. Some gamers may come to your game with context pre-installed, but you can't count on that.
ShippBoard Games is a board game design blog that updates most Mondays.