Okay for starters you should probably have edited your original question, not posted a new one, however..
Your page should probably contain the following:
1) A div that has an id, call it Values for simplicity and must also have attribute runat="server"
2) An ASP TextBox (which you already have), call this NewValueBox or something
3) An ASP Button (which you already have)
In the button click event do something like the following:
Session["Values"] = ((string)Session["Values"]) + NewValueBox.Text + "<br />";
Values.InnerHtml += ((string)Session["Values"]);
NewValueBox.Text = "";
The following is working Html and C# code I have just put together:
<body>
<form id="form1" runat="server">
<div id="ValuesDiv" runat="server">
</div><br />
<asp:TextBox ID="NewValueBox" runat="server"></asp:TextBox><br />
<asp:Button ID="Button1"
runat="server" Text="Add Value" onclick="Button1_Click" />
</form>
</body>
protected void Button1_Click(object sender, EventArgs e)
{
try
{
Session["Values"] = ((string)Session["Values"]) + NewValueBox.Text + "<br />";
}
catch
{
Session["Values"] = NewValueBox.Text + "<br />";
}
try
{
ValuesDiv.InnerHtml = ((string)Session["Values"]);
NewValueBox.Text = "";
}
catch
{
}
}