For road bike tires I often need at least one tire lever to remove it. I can almost always re-install a tire without a lever. With fatter MTB and touring tires, I usually don't need levers.

Some tire/wheel combinations can be very difficult on road bikes. If you find it hard to install a tire, try a different brand next time. If that doesn't work, it's either your technique or the rims.

BTW: If you find yourself needing a tire tool but don't have one, look at the lever on the wheel's quick release......