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'...
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.
There are even options for mounted ones.
(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 ...
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 ...
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 ...