You won't be able to see the tiles, but if you add additional lods to the map when you create it you'll be able to zoom in past the basemap tiles.
I was able to do it with this, by adding a few other additional levels of detail, you can add whatever scale and resolution you would like.
var customLods = [];
/* Custom LODs */
var mylods = [{
"level": 0,
"resolution": 156543.03392800014,
"scale": 5.91657527591555E8
}, {
"level": 1,
"resolution": 78271.51696399994,
"scale": 2.95828763795777E8
}, {
"level": 2,
"resolution": 39135.75848200009,
"scale": 1.47914381897889E8
}, {
"level": 3,
"resolution": 19567.87924099992,
"scale": 7.3957190948944E7
}, {
"level": 4,
"resolution": 9783.93962049996,
"scale": 3.6978595474472E7
}, {
"level": 5,
"resolution": 4891.96981024998,
"scale": 1.8489297737236E7
}, {
"level": 6,
"resolution": 2445.98490512499,
"scale": 9244648.868618
}, {
"level": 7,
"resolution": 1222.992452562495,
"scale": 4622324.434309
}, {
"level": 8,
"resolution": 611.4962262813797,
"scale": 2311162.217155
}, {
"level": 9,
"resolution": 305.74811314055756,
"scale": 1155581.108577
}, {
"level": 10,
"resolution": 152.87405657041106,
"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": 0.5971642835598172,
"scale": 2256.994353
}, {
"level": 19,
"resolution": 0.29858214164761665,
"scale": 1128.497176
}, {
"level": 20,
"resolution": 0.15,
"scale": 700
}, {
"level": 21,
"resolution": 0.1,
"scale": 500
}, {
"level": 22,
"resolution": 0.05,
"scale": 250
}, {
"level": 23,
"resolution": 0.01,
"scale": 100
}];
map = new Map("mapDiv", {
basemap: "streets",
center: [-120.327, 46.38],
zoom: 10,
lods: mylods
});
