You are attempting to reimplement the wheel by avoiding standard headers?
I assure you you will gain no benefit except learning how to implement your own stack class, and the result will most likely be lower performance, lower quality, not testings etc. of std::stack which will make no purpose for general usage.
You should get rid of the feeling to not include needed headers, and to do it all yourself.
people usually reimplement the wheel when they need more functionality specific for their project. but given this example there is not such reason.