Problem with the 74160 decade counter
After looking at the datasheet, the part normally pulls 60-70mA at 5V, dissipates 350mW.  This will be warm to the touch, not hot. Not sure why you're using an old TTL device, newer fast CMOS parts have standby currents in the 100uA range.

You mention adding a resistor to gnd, what's the other end of the resistor connected to? 

You may float TTL inputs, hopefully there are no DC loads on the outputs, otherwise the chip will draw current unnecessarily. 

Try this: connect a series resistor (5 ohms) inline with Vcc.  If you get a voltage drop > 0.5V across the resistor, then either the chip is damaged or the outputs (including RCO) are driving heavy loads.