An TextInput component for React Native that auto scales the height itself as the user types.

ExpandingTextInput.jsx

var React = require('react-native')
var {
    TextInput,
    StyleSheet
} = React;

var RCTUIManager   = require('NativeModules').UIManager;
var findNodeHandle = require('findNodeHandle');
var _ = require('lodash');

var ExpandingTextInput = React.createClass({
    propTypes: _.assign(React.TextInput.propTypes, {
        minHeight: React.PropTypes.number,
        maxHeight: React.PropTypes.number
    }),

    getDefaultProps: function () {
        return {
            minHeight: 45,
            maxHeight: Infinity,
            placeholder: 'Enter Text...',
            onChangeText: function () {},
        };
    },

    getInitialState: function() {
        return {
            height: this.props.minHeight
        };
    },

    onMeasureTextHeight: function(height) {
        if (this.isMounted()) {
            var minH = this.props.minHeight;
            var maxH = this.props.maxHeight;

            // it's not clear to me why this is needed but it seems to help
            // the height looks good at 50 and Xcode says it's 38
            height += 12;

            if (height < minH)  height = minH;
            if (height > maxH) height = maxH;

            if (this.state.height !== height) {
                this.setState({height: height});
            }
        }
    },

    resetHeight: function() {
        RCTUIManager.measureTextHeight(
            findNodeHandle(this.refs.input.refs.input),
            this.onMeasureTextHeight
        );
    },

    onChangeText: function(text) {
        this.resetHeight();
        this.props.onChangeText(text);
    },

    render: function() {
        var {style, onChangeText, multiline, ...others} = this.props;

        return (
            <TextInput
                ref = 'input'
                onChangeText = {this.onChangeText}
                multiline = {true}
                style={[defaultStyles.input, style, {height: this.state.height}]}
                {...others}
            />
        );
    }
});

var defaultStyles = StyleSheet.create({
    input: {
        fontSize: 12
    }
});

module.exports = ExpandingTextInput;

RCTUIManager+TextView.m

//
//  RCTUIManager+TextView.m
//  Tasker
//
//  Created by Brian Leonard on 7/31/15.
//

#import <UIKit/UIKit.h>
#import "RCTUIManager.h"
#import "RCTSparseArray.h"
#import "UIView+React.h"


@interface RCTUIManager (TextView)

@end

@implementation RCTUIManager (TextView)

/**
 * Returns information about the content inside of a RCTTextView
 */
RCT_EXPORT_METHOD(measureTextHeight:(nonnull NSNumber *)reactTag
                  callback:(RCTResponseSenderBlock)callback)
{
  if (!callback) {
    RCTLogError(@"Called measure with no callback");
    return;
  }

  [self addUIBlock:^(__unused RCTUIManager *uiManager, RCTSparseArray *viewRegistry) {
    UIView *view = viewRegistry[reactTag];
    if (!view) {
      RCTLogError(@"measureTextContent cannot find view with tag #%@", reactTag);
      return;
    }

    UIView *rootView = view;
    while (rootView && ![rootView isReactRootView]) {
      rootView = rootView.superview;
    }

    UITextView *textView = nil;
    NSArray *subviews = [view subviews];

    for(UIView *subview in subviews) {
      if ([subview isKindOfClass:[UITextView class]]) {
        UITextView * test = (UITextView *)subview;
        if (![test isHidden]) { // placeholder is hidden
          textView = test;
          break;
        }
      }
    }

    if (!textView) {
      RCTLogError(@"measureTextContent cannot find UITextView from tag #%@", reactTag);
      return;
    }

    // http://stackoverflow.com/questions/19046969/uitextview-content-size-different-in-ios7
    CGFloat measuredHeight;
    if (floor(NSFoundationVersionNumber) > NSFoundationVersionNumber_iOS_6_1)
    {
      // This is the code for iOS 7. contentSize no longer returns the correct value, so
      // we have to calculate it.
      //
      // This is partly borrowed from HPGrowingTextView, but I've replaced the
      // magic fudge factors with the calculated values (having worked out where
      // they came from)

      CGRect frame = textView.bounds;

      // Take account of the padding added around the text.

      UIEdgeInsets textContainerInsets = textView.textContainerInset;
      UIEdgeInsets contentInsets = textView.contentInset;

      CGFloat leftRightPadding = textContainerInsets.left + textContainerInsets.right + textView.textContainer.lineFragmentPadding * 2 + contentInsets.left + contentInsets.right;
      CGFloat topBottomPadding = textContainerInsets.top + textContainerInsets.bottom + contentInsets.top + contentInsets.bottom;

      frame.size.width -= leftRightPadding;
      frame.size.height -= topBottomPadding;

      NSString *textToMeasure = textView.text;
      if ([textToMeasure hasSuffix:@"\n"])
      {
        textToMeasure = [NSString stringWithFormat:@"%@-", textView.text];
      }

      // NSString class method: boundingRectWithSize:options:attributes:context is
      // available only on ios7.0 sdk.

      NSMutableParagraphStyle *paragraphStyle = [[NSMutableParagraphStyle alloc] init];
      [paragraphStyle setLineBreakMode:NSLineBreakByWordWrapping];

      NSDictionary *attributes = @{ NSFontAttributeName: textView.font, NSParagraphStyleAttributeName : paragraphStyle };

      CGRect size = [textToMeasure boundingRectWithSize:CGSizeMake(CGRectGetWidth(frame), MAXFLOAT)
                                                options:NSStringDrawingUsesLineFragmentOrigin
                                             attributes:attributes
                                                context:nil];

      measuredHeight = ceilf(CGRectGetHeight(size) + topBottomPadding);
    }
    else
    {
      measuredHeight = textView.contentSize.height;
    }

    callback(@[ @(measuredHeight) ]);
  }];
}

@end

This code is based on this answer, and made modifications so that it's easier to use.
Just treat it as a normal TextInput component with additional props, minHeight, and maxHeight.