Looking at the site, they give the option to buy a limited express ticket; you do not have to buy it if you do not want. And you can always by the Limited Express ticket later if you change your mind.
That being said, you're only going to benefit from this if you stay in Nikko for the entire period of the pass and use it aggressively.
If you wish to get to Nikko on regular trains only, probably the best route would be:
- Asakusa Station Tobu Isesaki (Tokyo Skytree) Line to Minami-Kurihashi
- Tobu Dobutsu Koen station If you got on a train to Minami Kurihashi, then you can stay on since it becomes the Tobu Nikko Line. Any other destination, get off and transfer to Tobu Nikko Line (Section Express) to Minami-Kurihashi
- Minami-Kurihashi station Tobu Nikko Line to Shin-Tochigi (if you're lucky, there may be one to Tobu-Nikko, but more likely this will be faster)
- Shin-Tochigi station Tobu Nikko Line to Tobu-Nikko.
- Arrive at Tobu-Nikko
On average, this route will have you transferring 2 times, and cost 1360 yen. This is a little over a half-hour slower than taking the paid Revaty, but note that I arbitrarily set the time to the first train, and the amount of time and/or transfers will change depending on the time you start (later in the day shows an hour difference between regular trains and Revaty). A Revaty ticket from Asakusa would be 2800 yen, but you are guaranteed a seat and will not need to transfer.
There are two stations, Nikko (JR) and Tobu-Nikko (Tobu). When traveling from the Tokyo area, most routes (both quick and cheap) will take you to Tobu-Nikko station. They are adjacent, though, so it doesn't matter.
Not all (limited) express trains cost extra. The trains that cost extra are luxury, and include reserved seating and food service (extra), such as the Revaty.
I've never used HyperDia before, but their English is a bit deceiving. The "Limited" and "Express" checkboxes are actually "Limited (paid)" and "Express (paid)". You should uncheck all checkboxes under "Route" except for "Ordinary Train" and "Walk" to get a route that doesn't cost any more than the basic fares.