plugins/contact-form-7/includes/classes.php
I don't know what version you have, so I can't say whether or not to just replace your file. Look for the parts that have my name (senica) in it.
Then in contact-form-7 create a new form but put your form in an hidden div like so:
<div class="student-request-hidden student-info" style="display:none">form here</div>
Then, when some clicks a button or something and you want to add the dynamic fields you have to use javascript to clone the hidden div, add it to your page and show it. Each field will have to have the "name" attribute with an incremental number.
For example: <input type="text" name="user-0" /><input type="text" name="user-1" />
Than the script should process them.
Hope this helps.