What will happen if I go through menopause with no hormone replacement?

Potential problems.. Every cell in a woman's body has estrogen and Progesterone receptors, so the entire body suffers once in menopause (when estrogen and Progesterone production cease completely). Symptoms range from depression, insomnia, hot flashes, fatigue, brain fog, to vaginal dryness, painful intercourse, low libido, dry skin, urine leakage, to increased risk of osteoporosis, dementia, and heart disease.
Possibly very little. Many women go through menopause with minimal problems and without hormones. Hormone therapy is only indicated for relief of symptoms and there are no proven medical benefits. The most common postmenopausal complaint I see is vaginal dryness which responds very well to topical estrogen (which is very low dose and not to be confused with "hormone therapy".).

Related Questions

If you are going through menopause can you have hormonal headaches almost daily if aren't taking hormone replacement therapy.

Possible. Different women have different symptoms at menopause and headaches are not uncommon. You should consult your gynecologist as she/he may have advice or medication. For good health - Have a diet rich in fresh vegetables, fruits, whole grains, milk and milk products, nuts, beans, legumes, lentils and small amounts of lean meats. Avoid saturated fats. Drink enough water daily, so that your urine is mostly colorless. Exercise at least 150 minutes/week and increase the intensity of exercise gradually. Do not use tobacco, alcohol, weed or street drugs in any form. Practice safe sex, if you have sex.