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.