Solution for expo SDK 16 and RN 0.43.2, TextInput Multiline={true} issue, In RN 0.46 fixed android onContentSizeChange issue.

AutoExpandingInput.tsx

import { action } from 'mobx';
import React, { PureComponent } from 'react';
import { StyleSheet, TextInputProperties, TextStyle, ViewStyle } from 'react-native';
import MultilineInput from './MultilineInput';

interface State {
  height: number;
}

// In RN 0.46 fixed conContentSizeChange problem.
export default class AutoExpandingInput extends PureComponent<TextInputProperties, State> {

  _inputRef;

  constructor(props) {
    super(props);

    this.state = {
      height: 0,
    };
  }

  @action
  componentWillReceiveProps(props) {

    if (!props.value) {
      this.setState({ height: 100 });
    }
  }

  render() {

    const {
      onContentSizeChange,
      style,
      ...otherProps,
    } = this.props;

    return (
      <MultilineInput
        {...otherProps}
        ref={this._ref}
        onContentSizeChange={this._onContentSizeChange}
        style={[styles.textInputStyle, style, { height: this.state.height }]}
      />
    );
  }

  focus = () => {
    this._inputRef.focus();
  }

  private _onContentSizeChange = (event): void => {

    const height = event.nativeEvent.contentSize.height;
    const { onContentSizeChange } = this.props;

    this.setState({ height });

    if (onContentSizeChange) {
      onContentSizeChange(event);
    }
  }

  private _ref = ref => {
    this._inputRef = ref;
  }
}

// set textAlignVertical: 'top' make sure text one the top and
// not shock when TextInput's height changed.
interface Styles {
  textInputStyle: TextStyle & ViewStyle;
}

const styles = StyleSheet.create<Styles>({
  textInputStyle: {
    fontSize         : 20,
    paddingHorizontal: 10,
    paddingVertical  : 10,
    textAlignVertical: 'top',
  }
});

MultilineInput.tsx

import React, { PureComponent } from 'react';
import { TextInput, TextInputProperties } from 'react-native';

interface State {
  cursorPosition: {
    end: number,
    start: number,
  };
  isSubmit: boolean;
  value: string;
}

/*
 * below only test on android, when set TextInput multiline={true}, some
 * keyboard will blur and call onSubmitEditing when press enter key,
 * so set blurOnSubmit={false}, and in onSubmitEditing() call onChangeText()
 * then add one line below value, on android onSubmitEditing will trigger twice,
 * so check if SubmitEditing already called.
 */
export default class MultilineInput extends PureComponent<TextInputProperties, State> {

  constructor(props) {
    super(props);

    this.state = {
      cursorPosition: { end: 0, start: 0 },
      isSubmit      : false,
      value         : '',
    };
  }

  render() {

    const {
      blurOnSubmit = false,
      multiline = true,
      onChangeText = this._onChangeText,
      onSubmitEditing,
      value = this.state.value,
      ...other,
    } = this.props;

    const { end, start } = this.state.cursorPosition;

    return (
      <TextInput
        {...other}
        blurOnSubmit={blurOnSubmit}
        multiline={multiline}
        selection={{ start, end }}
        onChangeText={onChangeText}
        onSubmitEditing={this._onSubmitEditing}
        onSelectionChange={this._onSelectionChange}
        value={value}
      />
    );
  }

  private _onChangeText = (value: string): void => {
    this.setState({ value });
  }

  private _onSelectionChange = (event): void => {
    this.setState({ cursorPosition: event.nativeEvent.selection });
  }

  // on android onSubmitEditing trigger twice, not test on ios.
  // when first time, add new line after value(thanks to nikolay-radkov),
  // second move cursor to next position (start of new line).
  private _onSubmitEditing = (event): void => {

    const { onChangeText, onSubmitEditing, value = this.state.value } = this.props;

    if (!this.state.isSubmit) {

      let newValue = value;

      if (value.length === this.state.cursorPosition.end) {
        newValue = value + '\n';
      } else {

        const ar = newValue.split('');

        ar.splice(this.state.cursorPosition.end, 0, '\n');
        newValue = ar.join('');
      }

      if (onChangeText) {
        onChangeText(newValue);
      } else {
        this._onChangeText(newValue);
      }

      if (onSubmitEditing) {
        onSubmitEditing(event);
      }

      const newPosition = this.state.cursorPosition.end + 1;

      this.setState({
        cursorPosition: { start: newPosition, end: newPosition },
      });
    }

    this.setState({ isSubmit: !this.state.isSubmit });
  }
}

I got an error when trying to use it:
Undefined isn't an object evaluation _react.PorpTypes.string