Children under 2 are free in the Highfield Discovery Garden at Glenwood Gardens. This is the only area with a per-person fee. Admission for the rest of the park is free for everyone with the Hamilton County Great Parks tag on their vehicle.
Besides a Hamilton co. Park pass, You may be talking about the nature corner area, I believe there is an admittance fee for that, but I've not been in there. Not sure of the price.
I believe park admittance is free, but you must have a motor vehicle permit to bring in your car. That's $10 annually, or $3 for a single day. There is no per-person cost regardless of age.
I'm not an employee, but I'm pretty sure it's free for kids under 2.
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