Using own basemap with ArcGIS API for Javascript?

The API Reference for the Map Class says the following:

The following are valid options: "streets" , "satellite" , "hybrid", "topo", "gray", "oceans", "national-geographic", "osm". As of version 3.3

You should instead, just define a new ArcGIS Tiled Map Service Layer and use it as follows:

var baseMapLayer = new esri.layers.ArcGISTiledMapServiceLayer("your URL");
map.addLayer(baseMapLayer);

For arcgis javascript SDK version 3.x you can use esri/basemaps class to declare and register custom basemaps and use with your map: https://developers.arcgis.com/javascript/3/jsapi/esri.basemaps-amd.html

Declare and register custom basemap:

Basemaps.mybasemap = {
  title: 'My custom basemap',
  thumbnailUrl: 'https://js.arcgis.com/3.22/esri/images/basemap/satellite.jpg',
  //itemId: 'ulas',
  baseMapLayers: [
    { url: "https://services.arcgisonline.com/ArcGIS/rest/services/Specialty/DeLorme_World_Base_Map/MapServer" }
  ]
};

Use custom basemap in map constructor:

var map = new Map("mapDiv", {
  basemap: "mybasemap",
  center: [-122.69, 45.52],
  zoom: 3
});

Or you can apply your custom basemap to existing map:

map.setBasemap("mybasemap");

Full code example: https://codepen.io/digz6666/pen/wPwPbW

For arcgis javascript SDK version 4.x you can use esri/Basemap.

Declare base map layer and base map:

var baseLayer = new MapImageLayer({
  url: "https://services.arcgisonline.com/ArcGIS/rest/services/Specialty/DeLorme_World_Base_Map/MapServer"
});
var myBasemap = this.esri.basemap({
  baseLayers: [baseLayer],
  title: 'My custom basemap',
  id: 'my_custom_basemap'
});

Apply basemap to existing map object:

map.basemap = myBasemap;

You don't have to specify a built-in basemap. Add your custom basemap as a layer like so.

var map;
require(["esri/map", "esri/layers/ArcGISTiledMapServiceLayer", "dojo/domReady!"],
function (Map, ArcGISTiledMapServiceLayer ) {
    map = new Map("map", {                
        center: [-76.756, 40.241],
        zoom: 8
    });            
    var customBasemap = new ArcGISTiledMapServiceLayer(
    "http://services.arcgisonline.com/ArcGIS/rest/services/NatGeo_World_Map/MapServer");
    map.addLayer(customBasemap);
});