Common Compound Words:
* Handbag
* Handshake
* Hardback
* Headache
* Headband
* Headhunter
* Heartbeat
* Heartbreak
* Heathenism
* Heavyweight
* Highchair
* Highlighter
* Highway
* Homepage
* Horseback
* Houseboat
* Housewarming
* Housewife
* Hummingbird
* Hungerstrike
Less Common Compound Words:
* Hairpiece
* Halfback
* Halfhearted
* Handpicked
* Handstand
* Hard-boiled
* Hardship
* Hardwood
* Headfirst
* Headstrong
* Healthful
* Hemlock
* Hereafter
* Highborn
* Highland
* High-pitched
* High-ranking
* High-rise
* Highwayman
* Homebody
* Homestead
* Honeycomb
* Horsefly
* Horsepower
* Hotdog
* Hothead
* Houseplant
Phrases as Compound Words:
* Hand-me-down
* Heart-to-heart
* Hold-up
* Hot-tempered
This is not an exhaustive list, as there are many compound words out there!
Do you have a specific category or theme you're interested in for compound words starting with "h"? I'd be happy to help find more examples.