Correction:
obj.value[0].assets.logo["436209"].url = 'foo';
// or
var foo = obj.value[0].assets.logo["436209"].url;
This assumes that your object is well formed and continues with more parts of obj.value[0].
Specifically, if your object were completed, perhaps, like this:
var obj = {
"error": { "occured": "false" },
"errors": [],
"executiontime": 2500,
"metadata": {},
"value": [{
"activity_duration": "1 hour, ½ day & full day packages",
"adult_rate_high_period_high_price": 275,
"adult_rate_high_period_low_price": 49,
"adult_rate_low_period_high_price": "",
"adult_rate_low_period_low_price": "",
"amenities": [],
"assets": {
"logo": {
"436209": {
"asset_type": "image",
"caption": "",
"credit": "",
"description": "",
"exists": "true",
"height": 82,
"label": "Copy of Monarch logo",
"latitude": 0,
"longitude": 0,
"market": "$",
"o_id": 3221685,
"type_o_id": 2543991,
"unique_id": 436209,
"url": "http://c0481729.cdn2.cloudfiles.rackspacecloud.com/p-DD951E3E-C7AF-F22C-77E98D299833B38F-2544001.jpg",
"width": 220
}
}
}
}]
};