app.rb

require 'sinatra'
enable :sessions

get '/' do
    session[:words] ||= []
    session[:words].push params[:word] if params[:word]
    erb :view
end

view.erb

<form>
    <ul>
        <% session[:words].each do |word| %>
            <li><%= word %></li>
        <% end %>
    </ul>
    <input type="text" name="word" /> <input type="submit" value="ok" />
</form>