Code

using UnityEngine;
using System.Collections;
using System.Collections.Generic;

class MyAnimController : MonoBehaviour {
    public Queue<int> queuedStateHashes = new Queue<int>();

    private int currentStateHash;

    public string defaultState;

    public Animator theAnimator;

    bool buttonPressed = false;
    int buttonNumber = 1;

    AnimatorStateInfo info;

    private void Awake() {
        currentStateHash = Animator.StringToHash(defaultState);
        //StartCoroutine(PlayStates());
    }

    //how you start and exit this is up to you, in this case,
    //I'll assume it starts on awake
    //private IEnumerator PlayStates() {
    //    while (true) {
    //        SetAnimStates();
    //
    //        yield return null;
    //    }
    //}

    private void SetAnimStates() {
        info = theAnimator.GetCurrentAnimatorStateInfo(0);

        if (info.normalizedTime >= 1 && queuedStateHashes.Count > 0) {
            currentStateHash = queuedStateHashes.Dequeue();
            print("starting");
            theAnimator.Play(currentStateHash, 0, 0);
        }
        else if (queuedStateHashes.Count == 0) {
            //queuedStateHashes.Enqueue(currentStateHash);
        }

        //print(queuedStateHashes.Count);
    }

    private void Update() {
        if (buttonPressed) {

            buttonPressed = false;

            switch (buttonNumber) {
                case 1: queuedStateHashes.Enqueue(Animator.StringToHash("Anim4")); break;
                case 2: queuedStateHashes.Enqueue(Animator.StringToHash("Anim4")); break;
                case 3: queuedStateHashes.Enqueue(Animator.StringToHash("Anim5"));
                        queuedStateHashes.Enqueue(Animator.StringToHash("Anim6")); break;
                case 4: queuedStateHashes.Enqueue(Animator.StringToHash("Anim8")); break;
                case 5: queuedStateHashes.Enqueue(Animator.StringToHash("Anim7")); break;
                default: queuedStateHashes.Enqueue(Animator.StringToHash(defaultState)); break;
            }

            //queuedStateHashes.Enqueue(stateHash);
        }
        SetAnimStates();
    }

    public void pressButton(int number) {
        buttonPressed = true;
        buttonNumber = number;
    }
}