I tend to get better help here then the ESRI forum so...
Using http://developers.arcgis.com/en/javascript/sandbox/sandbox.html I was able to create a simple application which demonstrates that if you set a non-trivial lods on a map the basemap will not behave correctly (try zooming beyond level 15).
Any workarounds?
<!DOCTYPE html>
<html>
<head>
<title>Create a Map</title>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
<meta http-equiv="X-UA-Compatible" content="IE=7, IE=9, IE=10">
<link rel="stylesheet" href="http://serverapi.arcgisonline.com/jsapi/arcgis/3.4/js/dojo/dijit/themes/claro/claro.css">
<link rel="stylesheet" href="http://serverapi.arcgisonline.com/jsapi/arcgis/3.4/js/esri/css/esri.css">
<style>
html, body, #mapDiv {
padding: 0;
margin: 0;
height: 100%;
}
</style>
<script src="http://serverapi.arcgisonline.com/jsapi/arcgis/3.4/"></script>
<script>
require([
"esri/map",
"esri/layers/ArcGISTiledMapServiceLayer"
],
function (Map, ArcGISTiledMapServiceLayer) {
var webLods = [
//{ "level" : 0, "resolution" : 156543.033928, "scale" : 591657527.591555 },
//{ "level" : 1, "resolution" : 78271.5169639999, "scale" : 295828763.795777 },
//{ "level" : 2, "resolution" : 39135.7584820001, "scale" : 147914381.897889 },
//{ "level" : 3, "resolution" : 19567.8792409999, "scale" : 73957190.948944 },
//{ "level" : 4, "resolution" : 9783.93962049996, "scale" : 36978595.474472 },
//{ "level" : 5, "resolution" : 4891.96981024998, "scale" : 18489297.737236 },
{ "level" : 6, "resolution" : 2445.98490512499, "scale" : 9244648.868618 },
{ "level" : 7, "resolution" : 1222.99245256249, "scale" : 4622324.434309 },
{ "level" : 8, "resolution" : 611.49622628138, "scale" : 2311162.217155 },
{ "level" : 9, "resolution" : 305.748113140558, "scale" : 1155581.108577 },
{ "level" : 10, "resolution" : 152.874056570411, "scale" : 577790.554289 },
{ "level" : 11, "resolution" : 76.4370282850732, "scale" : 288895.277144 },
{ "level" : 12, "resolution" : 38.2185141425366, "scale" : 144447.638572 },
{ "level" : 13, "resolution" : 19.1092570712683, "scale" : 72223.819286 },
{ "level" : 14, "resolution" : 9.55462853563415, "scale" : 36111.909643 },
{ "level" : 15, "resolution" : 4.77731426794937, "scale" : 18055.954822 },
{ "level" : 16, "resolution" : 2.38865713397468, "scale" : 9027.977411 },
{ "level" : 17, "resolution" : 1.19432856685505, "scale" : 4513.988705 },
{ "level" : 18, "resolution" : 0.597164283559817, "scale" : 2256.994353 },
{ "level" : 19, "resolution" : 0.298582141647617, "scale" : 1128.497176 }
];
var map = new Map("mapDiv", {
center: [-85, 35],
zoom: 3,
basemap: "streets",
lods : webLods,
slider : true,
sliderStyle : "large"
});
});
</script>
</head>
<body class="claro">
<div id="mapDiv"></div>
</body>
</html>