Years ago I had some physical therapy and the therapist would tell me to do a certain number of repetitions of an exercise, then, while I did them, he would try to have a conversation with me. I found it challenging to both maintain my side of the conversation and keep an accurate count in my head. Eventually I realized accuracy wasn't so important and the exact number of reps didn't matter, but by then I had developed a way to off-load counting to my hands, using one hand to count as high as 60. Here's how.
I'd heard somewhere that the ancient Babylonians may have used 12 so much in their number system because, instead of counting on the tips of all 10 fingers, they counted on the bones of the fingers of one hand (not including the thumb), so I adopted the technique. To count 1, I put the tip of my thumb against the bone at the end of my pointer finger, and to increase to 2 and 3 I slide my thumb onto bones progressively closer to the hand. At 4 I reset my thumb to the end bone of the middle finger, at 7 to end of the ring finger, and at 10 to the end of the pinky.
To count higher, I use the same pattern but with my thumb on different sides of the fingers. The "thumbward" side is 13 through 24. Counting 25 through 36 requires tucking my thumb under each finger, so the nail is against the bone. The backs of the fingers supply 37 through 48.
It seemed awkward to stop at 48, and a bit incongruous to find two positions of, say, my thumb to count up to 50, so I count 49 to 60 using the tips of each finger and their next two joints (not the knuckles). I've counted by placing the tip of my thumb against each tip or joint in turn, but it can be a slightly difficult to distinguish thumb-against-joint from thumb-against-bone, so I've also tried curling each finger to place its tip or joints against the joint of the thumb instead of its end. Either way is different enough to require some additional concentration.
I do have relatively long fingers with decent flexibility, so I don't know if smaller hands or limited dexterity could make counting this way hard.
Besides counting reps for exercises that don't require hands, I've also counted paces when hiking to help gauge distance.
The only disadvantage of this system I've found so far is there's no longer much point to using the phrase "I can count the number of ______ on one hand."
If you have other trips for counting on your hands, feel free to email me.