I remember the first lecture on classical thermodynamics where our teacher asked if anyone knew what temperature actually is. At first you think this question is ridiculous, but then you realize you haven't the slightest idea. I found this to be a very pleasant moment in my education. Similarly, many other basic thermodynamics quantities are interesting to contemplate, with entropy definitely being one of them.
Also, with some linear algebra you can work out one of the most beautiful classical mechanics riddles I came across (presented to me by my classical mechanics teacher): take a frictionless 1D "plane" with a single wall. Place two masses $m_1$, $m_2$ on one side of the wall, and let $m_1$ travel at initial speed $v_1$ towards the stationary $m_2$ that is placed between $m_1$ and the wall. The masses will collide, sending $m_2$ towards the wall and rebounding back towards $m_1$. The super-gorgeous question is: how many times will the balls collide as a function of $m_1$, $m_2$, $v_1$? Collisions are completely elastic, no friction, all mechanical energy is conserved. It's not "counterintuitive", but it's a demonstration of great beauty in the work of precise mathematical tools in a physical context.
Finally, I think non-inertial reference frames present many counterintuitive scenarios, such as the Coriolis force, balloons in accelerating cars, pendulums, motion of icebergs and so on.
Additional rigid-body examples:
Suppose you have two barrels with the same mass rolling down a hill. One of them has the mass concentrated close to its major axis and the other has its mass more spread out. Which would roll down faster? more generally, what is the optimal mass distribution to make the barrel roll fastest?
You're driving a car uphill (without slipping, just driving normally). To which direction does friction force works upon your car?
And to wrap things up with one of most famous classical mechanics problems of all time: the brachistochrone: suppose you want to design a slope such that a skier that slides without friction and only due to gravity will reach the endpoint in the shortest time possible. What is the curve that achieves this goal? Rephrasing, what is the "fastest curve" you can make? Is it a straight line? A quarter-circle? How should one approach this problem?