How can I easily keep track of my landing count during pattern work?

I asked my CFI this after a day where we did over a dozen landings. He showed me his kneeboard notebook - there was a neat little row of tally marks next to the METAR info for each field we'd landed at.

The trick he used was that he was marking takeoffs, because ultimately what takes off must land (note; this won't work for spaceplanes). During the climb you're really just holding it steady and listening for calls; I find in a c172 that's a good time to grab the pen and make a mark on the kneeboard.

Pace count beads, which are just some beads on a rope that doesn't let them readily move, would work well if you want to keep track while in the air.

I use my ADF. Set the frequency to 1000 when I start, bump it by one for each landing.

(Might as well use it for SOMETHING!)

Get a Tally Counter, and push the button once per landing. No batteries, nothing to fail, easy to use. Not much more you could ask for.

https://tallycounterstore.com/finger-tally-counter-quantity-discounts/

There are even options for mounted ones.

https://tallycounterstore.com/mounted-tally-counter/

(Disclaimer: I'm not a pro at all, I don't even have any license yet)

If you do indeed have a smartphone with you, you might consider just letting a voice recording run for the duration of your pattern work and call out your landings. When you're back on the ground, you can just listen through the recording and count the landings.

This might also have another interesting use, as you could actually comment on your maneuvers, i.e. if you're happy or unhappy with something, so you could analyse it later.

Afaik, Flight Chops (the aviation youtuber) initially started recording his flights on camera in order to be able to analyse them later, so that might be a way as well.

(Also, welcome to aviation.stackexchange ?)

The other answers have provided some easy ways to increment your count without too much distraction, but if you are really "prone to forget [...] when things get busy" as you say, this might not be good enough. I would therefore recommend a solution which does not require any action on your side.

Most smartphones today have a pressure sensor. There are apps that can record the pressure data over time (e.g. this one). All you have to do is start the recording before you go flying and then later look at the pressure vs. time plot. In an un-pressurized aircraft it will be easy to see your landings (these sensors are typically accurate enough to tell the difference between your feet and your head).

Alternatively, you could record your GPS position over time (e.g. this app) and look at your flights that way. Note that both of these options would work even with the phone in flight mode.

Almost every calculator can be converted into counter just by typing something like 1 + = (and then every time you press = the value is incremented).

This can be used if you need aircraft instruments for the training itself instead.

The trick my CFI taught me is to use your Course Deviation Indicator or ADF to keep track of landings. After your first landing, bug a course of 010. After your second landing, bug 020. You can reach over and adjust the knob after every landing without having to juggle a pen and a notebook. It's still a manual step, though.

You could also use a product like Foreflight or CloudAhoy or even a service like ADSBExchange to count the number of approaches you made after the fact.