Single chip to control inputs and outputs?

I'm working on a project that's quickly pushing past the number of pins on my Arduino. I need to read 7 input pins and control 15ish LEDs (PWM not required). Is there any sort of chip that can multiplex both inputs and outputs and be controlled through a single digital interface?

Thanks!

Sure - 74AC299PC, universal parallel load shift-in/shift-out register.
Use one to read in your inputs, 2 to send out your outputs.
56 cents at Newark.com.
On the input side, if you don't want to sample the inputs (parallel load the state of all 7 at once), then you can run them thru a multplexer such as CD4051, lets you open a channel between the source and your input pin, you can read the same pin without the shift register delays.
If the analog nature of the switch is problem, then strictly digital versions exist also.
Can suggest alternatives with more details on your signals, analog vs digital, switching speeds, etc.

Thanks. I'll look into that one.

I'm not sure what you mean by sampling the inputs.

I need to control an LCD (roughly 12 pins, 10 digital, 2 analog), some sort of sound device (probably an Adafruit Wave Sheild) which is around 9 digital pins I believe, around 10 LEDs (just on or off), and a keypad which requires 7 digital inputs to read.

I'm starting to think I would be better off just going for an Arduino Mega...

Sampling - think of the way a shift register works. You have 8 registers there, 1 clock signal. So the clock gives you a sample of what's on the lines when the clock triggers the registers.

Now that you've described your needs better, better recommandations can be made.

Mega is sounding good, you can implement a lot of the libraries (keypad, LCD, etc) without having to fool around adapting hardware to them.
Shift register for the LEDs is still good, the 74AC299 can control a lot of current for the LEDs without cutting into the Mega current limits.
Or, a couple of ULN2803s for direct drive if you've got the pin count.

You've got 38 accounted for out of the 70+ available.