For a private home health aide, what do others' give in addition to basic compensation?

If this is a private hire caregiver cash is always great as a bonus. Or a gift card to a place you know they shop
If it is from an agency the agency may have restrictions as to the amount an employee can accept.
Do not give more than you can afford to give. (Or if your loved one is paying the caregivers do not give more than they can afford to give) Gifting can be tricky if there is ever a "look back".

CaringDaughter7, back when my Dad had 3 shifts of caregivers at home, those who were scheduled regularly I gave them at Christmas a $25 to $50 gift card from Target. I first checked with their Agency, and the Agency said it was ok.

When my Dad moved into senior living, Dad brought along 2 of the caregivers [one during the week, the other weekends] to work their 1st shift, with the approval of the facility. Once Dad had passed on, I sent them both a large bonus check for all the wonderful work, Dad really enjoyed having them around. And it gave me peace of mind.

They always appreciate the cash gifts at Christmas! This is a little over the top but after Dad died, we kept our health care aide for Mom, who was getting sicker. Mom wanted to go on a cruise before she couldn't go any more, and she needed the aide to help her in the bathroom. The only way to do this was to take the aide along, so the aide got a cruise and was very grateful. She was also able to see and spend time with relatives in one of the ports. It took the aide, a friend, and me to handle Mom, who was in a wheelchair, on the cruise. Nearly wore all of us out, but Mom was happy. We also included the aide in family holidays and she was not expected to do any extra work, just look after the parent. It's good to be thoughtful to them whenever possible because they do work so hard.