Apparition/Disapparition is the first thing that comes to mind. It requires the three D's: Destination, Determination, and Deliberation.
Borrowing this quote from Vishvesh, sourced from the books:
He waved his wand. Old-fashioned wooden hoops instantly appeared on the floor in from of every student.
“The important things to remember when Apparating are the three D’s!” said Twycross. “Destination, Determination, Deliberation!
“Step one: fix your mind firmly upon the desired destination,” said Twycross. “In this case, the interior of your hoop. Kindly concentrate upon that destination now.”
“Step two,” said Twycross, “focus your determination to occupy the visualised space! Let your yearning to enter it flood from your mind to every particle of your body! “
“Step three,” called Twycross, “only when I give the command... turn on the spot, feeling your way into nothingness, moving with deliberation. On my command, now... one —”
Harry Potter and the Half-Blood Prince, Chapter 18, Birthday Surprises
Another one that comes to mind is the Patronus Charm, which requires a particularly strong happy memory and intent for the desired outcome:
Harry Potter: “And how do you conjure it?”
Remus Lupin: “With an incantation, which will work only if you are concentrating, with all your might, on a single, very happy memory.”
Harry Potter and the Prisoner of Azkaban
As you mentioned in the comments below, Riddikulus is also a candidate in that it is directed towards a specific target, but the requirement seems along the lines of what it takes to cast Expecto Patronum.
So it does seem that there are other spells that very much require strong intent to accomplish, but nothing that seems to come close to needing the same degree of intent as the unforgivables (really wanting and enjoying their outcomes, or else they fail).
Spells in general seem to require varying degrees of the following three components: wand work, incantation, and intent. Some spells require all three, whereas others can leave out one or two (wandless and non-verbal), which makes the third much more important. Relating to the example above, disapparation doesn't seem to need a wand or verbal component, but requires strong intent. The unforgivables in particular, being as a strong as they are, seem to require a great deal of all three (although there might be an exception wherein strong enough intent negates the need for the verbal component).