I would characterise a game's depth as its capacity to engage a player beyond their first session. To nuance this further, depth implies a long-term engagement with game systems (as opposed to purely content, although this can get a little fuzzy).
I've discussed why this matters commercially here - greater depth allows for greater lifetime player value as well as heightened network effects. But it's also frequently a creative goal, with designers discussing "elegance" in terms of emergent depth from simple rulesets.
When you're working on design or evaluating a game, you can think about depth in the following dimensions:
Decision Space
What is the player thinking about as they make their next move?
How about the next time they get to work on their character, or choose the next stage in the tech tree?
If games are a "series of interesting decisions", can we articulate what should be going through a player's head as they make a choice and enumerate all of the dimensions they are considering? If they are thinking about how to realise their own unique build, which weapon is coming up next that they might pick up, some element of chance they are banking on as well as how to dodge the current enemy's attack then we could be going in the right direction.
Horizons
Time is fundamental to games.
Yes, a game will be functioning well if the immediate interactions are fun and lively but does the player know where they are going and why they are going there? If they beat the game with a certain style, what might they want to try next time around? If they are still playing the same run in 10 years…why might that be?
Trivial decisions with zero long-term consequences are often the enemy of depth - aim to create layering over time. Repetitiveness or lack of variety can dull excitement, so think about cadence over variable timeframes when planning out your content.
Mastery
Many players like to work towards mastery. In some games this is purely about mechanical skill or some combination of skill with competitive psychology. In others, it's about strategic or systemic optimisation.
The question to ask about your game is "what does mastery feel like"? If you can create steps along the way which are satisfying to achieve and an ultimate goal which seems almost insurmountable, then you'll have designed a journey which players want to take.
Social
We've seen a huge rise of co-op games recently, from Split Fiction to Murky Divers - adding a social dimension to traditionally solo game experiences has a profound impact on depth. As soon as another human being is in the mix, whether cooperatively or competitively, things get positively complicated.
Creativity and Systems
Many successful indie games stray into becoming simulations or even tools. You will have a huge advantage if players can use your game to realise an abstract idea or fantasy - we're seeing this at the moment with the huge popularity of Schedule 1 which invites players to progress "from small-time dope pusher to kingpin". Minecraft didn't blow up simply because of its charming voxel landscapes or survival mechanics - it offered players the chance to shape their own world.
Mystery
We commonly think about depth in terms of mechanics but there are narrative and conceptual elements which play into it as well - mystery is perhaps the most intriguing of these. From the meta-complexities of Inscryption to the shifting geometries of Blue Prince, keeping players chasing answers as well as abstract goals can be a potent combination.
Deep Dive
SuperBunnyHop's excellent recent video about the origins of strategy games reinforced to me that, for as long as humans have been designing games, they've been looking for ways to model the world and capture its details at a high resolution.
But depth doesn't necessarily always emerge from a huge number of rules or vast array of features - it is fundamentally about creating broad possibility spaces and intense player engagement. Designers are often cautious and pessimistic, focussing too narrowly on game feel, approachability and moment-to-moment interactions: stepping back to think in terms of years rather than minutes can lead in some fruitful directions.
Can depth be proven or demonstrated? In the current marketplace, if you show up to a publisher with a pitch that already contains metrics around both retention and engagement, evidence that you understand player pain points and how to eliminate them and a clear idea of player aspirations within your game, then you are going to be ahead of the pack by default.