React Native: Constraining Animated.Value

Use the extrapolate, extrapolateRight or extrapolateLeft properties:

const maxX = 200;
const constrainedX = this.state.pan.x.interpolate({
  inputRange: [0, maxX],
  outputRange: [0, maxX],
  extrapolateRight: 'clamp'
})

https://facebook.github.io/react-native/docs/animations.html#interpolation

The solution I came up with: Since Animated.event() returns a function, you can just pass your processed gestureState, with constrained values, into that function.
It'd look something like this:

onPanResponderMove: (evt, gestureState) => {
    let newdx = gestureState.dx, //DO SOMETHING WITH YOUR VALUES
        newdy = gestureState.dy;
    Animated.event([null,{
        dx: this.state.pan.x,
        dy: this.state.pan.y
    }])(evt, {dx: newdx, dy: newdy});
},

Whether it's it should be done is debatable though.

The solution provided by @stinodes will degrade performance. Why not using interpolate in the following way:

const maxX = 200;
const constrainedX = this.state.pan.x.interpolate({
  inputRange: [0, maxX, Infinity],
  outputRange: [0, maxX, maxX]
})

const constrainedX = this.state.pan.x.interpolate({
      inputRange: [leftLimit, 0, rightLimit],
      outputRange: [leftLimit, 0, rightLimit],
      extrapolate: 'clamp'
})

extapolate: 'clamp' will keep it in bounds.