According to the ArcGIS API for Python System requirements page lists only 1 requirement:
System requirements
In order to use version ArcGIS API for Python, you need Python 3.5 or later.
 
I wouldn't say that statement is incorrect, just incomplete.  Installing the ArcGIS API for Python in a clean Miniconda installation gives a better idea of requirements (I created a new arcgis virtual environment😞
C:\Users\bixb0012\Miniconda3>Scripts\Conda.exe install -n arcgis -c esri arcgis
The following NEW packages will be INSTALLED:
    arcgis:              1.3.0-py35hdae95b8_1     esri
    bleach:              2.1.2-py35_0
    certifi:             2017.11.5-py35h456c6ae_0
    colorama:            0.3.9-py35h32a752f_0
    decorator:           4.2.1-py35_0
    entrypoints:         0.2.3-py35hb91ced9_2
    html5lib:            1.0.1-py35h047fa9f_0
    ipykernel:           4.8.0-py35_0
    ipython:             6.2.1-py35h4a2ac14_1
    ipython_genutils:    0.2.0-py35ha709e79_0
    ipywidgets:          5.2.2-py35_1
    jedi:                0.11.1-py35_0
    jinja2:              2.10-py35hdf652bb_0
    jsonschema:          2.6.0-py35h27d56d3_0
    jupyter_client:      5.2.1-py35_0
    jupyter_core:        4.4.0-py35h629ba7f_0
    markupsafe:          1.0-py35hc253e08_1
    mistune:             0.8.3-py35_0
    nbconvert:           5.3.1-py35h98d6c46_0
    nbformat:            4.4.0-py35h908c9d9_0
    notebook:            5.3.1-py35_0
    numpy:               1.10.1-py35_0
    pandas:              0.20.3-py35he2ce742_2
    pandoc:              1.19.2.1-hb2460c7_1
    pandocfilters:       1.4.2-py35h978f723_1
    parso:               0.1.1-py35he39c48a_0
    pickleshare:         0.7.4-py35h2f9f535_0
    pip:                 9.0.1-py35h691316f_4
    prompt_toolkit:      1.0.15-py35h89c7cb4_0
    pygments:            2.2.0-py35h24c0941_0
    python:              3.5.3-5                  esri
    python-dateutil:     2.6.1-py35h6b299a3_1
    pytz:                2017.3-py35hb294b1a_0
    pyzmq:               16.0.3-py35h1861011_0
    send2trash:          1.4.2-py35_0
    setuptools:          38.4.0-py35_0
    simplegeneric:       0.8.1-py35h6406082_0
    six:                 1.11.0-py35hc1da2df_1
    testpath:            0.3.1-py35h06cf69e_0
    tornado:             4.5.3-py35_0
    traitlets:           4.3.2-py35h09b975b_0
    vc:                  14-h0510ff6_3
    vs2015_runtime:      14.0.25420-0             esri
    wcwidth:             0.1.7-py35h6e80d8a_0
    webencodings:        0.5.1-py35h5d527fb_1
    wheel:               0.30.0-py35h38a90bc_1
    widgetsnbextension:  1.2.6-py35_3             esri
    win_unicode_console: 0.5-py35h56988b5_0
    wincertstore:        0.2-py35hfebbdb8_0
    winkerberos:         0.6.0-py35_0
 
The ArcGIS API for Python (a.k.a. arcgis) has quite a few dependencies.  Fortunately, if you are using Conda to install it, the dependencies are taken care of for you.
 
Note, installing arcgis through Conda also installs Python 3.5.3-5.  The packaging of the ArcGIS API for Python has gotten tighter over time.  Each release is now developed against a specific version of Python, which makes the statement, "you need Python 3.5 or later", on the requirements page partially incorrect.
 
If you install the ArcGIS API for Python on a machine without ArcGIS Pro, you will not have access to ArcPy, so some of the functionality of the ArcGIS API for Python will not be available.