What are your favorite accessories?

I have a combination dice tower, dice try and dice storage system that I would fight tooth and nail to keep.

In storage mode. Exposing its guts. In active use.

That looks amazing! Did you make it yourself or where did you get it from?
- I got it from a shop on Taobao. (Along with the dice you see there including the skull-face D6s, the ancient Chinese d18, the Chinese Chess dice, ...)

Card sleeves. Not on all games, but the one you manipulate the cards a lot. My last sleeved game is Sea, Salt & Paper.
Card playing mat. Makes the card manipulation so much easier on flat tables.

Which mats and sleeves can you recommend?

We usually sleeve games once its obvious we are going to play it a lot. Terraforming Mars, Clank, Dominion, Ark Nova all come to mind. But sometimes the sleeves are rather sticky. It feels like good sleeves are super expensive.
- Dragon Shield Matte are amazing! IMO I haven't felt an experience of holding cards like Dragon Shield (and I've used a lot of other sleeves for games that don't fit the standard card size). Just my two cents.
- Not OP but Arcane Tinmen make pretty nice sleeves. I recommend the non-glare ones. For playmats I can recommend the 3’x3’ off-brand X-Wing mats.

Mine:

A couple of very cheap sectioned trays. Shallow enough to make grabbing tokens easy and enough sections to organize them. We use these for pretty much every game with tokens.
A side-table to help deal with table hogs.
A couple of those foldable dice trays. I don't actually use them for dice but as a flexible funnel when bagging things.
Notepad, pencils, eraser
An empty tea tin. I use this for some games where you frequently draw items from a bag because it's a lot faster.

We do have a 3D printer and I love improving my games with all sorts of little things. I did also print some 1-part, 2-part, 3-part and 4-part stackable, multicolored trays which I love and use for many game.

And I have also folded some origami for Everdell which is a great addition and still all fits into the box:

But my favorite is our Burgundy Hexagon - it holds all the components of the game and includes also two dice trays. I would not want to play it without it, it makes playing much more fluid. The downtime inbetween phases is super short.

And all put back together:

My favourite cheap accessory is silicone muffin cups. Available in multiple colours. Fits all your different resources. I think nearly every second person in our club has some. Extra nice: cups with a fold halfway up. Perfect to make half cups for fewer recourses. When not in use, make cupcakes.

My calico games bag that says, "I have too many boardgames, said no one ever" (c:

A large piece of orange felt as a table topper.

A low ceramic 4 compartment serving tray, for counters and other bits. It is the perfect dimension, weight and feel.

X-trays to store game pieces is life-changing. The lids hold on well and they are stackable. They come with me for any game. And they are like $1.50 each

Are those the trays used to grow plants?
- What? It’s for board game pieces

Pennygems. Absolutely kills me they aren't sold anymore.

Burger Tokens are pretty great.
- These look excellent, thank you! I'll probably be picking up a Keyforge set.

I like my neoprene mousemat-style playmat. Makes it easier to pick op cards and other components and offers slight protection as well.

I use a 36”x36” one made for X-Wing with a nice, low-key space theme.

Beer.

A thin piece of felt the size of my dining room table.

I use it every time we play anything, at minimum, weekly.