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There are no countries that ban the US dollar, because this is a global trading currency. It is a currency that controls the effective rates of other currencies due to the large impact of the US economy on other global economies.
In addition, a large majority of international money movement takes places in so called "major currencies" such as the US dollar, the Euro, and the UK pound. Therefore, no matter how you start out with, when exchanging funds your money there is a high probably that is is being converted to USD and (the equivalent) is being transferred or exchanged.
You should always attempt to trade in the local currency, and try to have minimal local currency overhead when you exit the country (unless you plan on frequent visits to that destination). Converting local currency back can be more expensive than changing to the currency; however you can get some reprieve from this double-hit as some money changers offer a guaranteed buyback rate - but only within a couple of days of the initial transaction.
There are some strange local issues though. Some places do not accept the $100 bill because it is the one most often forged, others will only accept new bills and refuse old (used) notes.
In all cases, you are guaranteed the following:
Major hotel chains will gladly accept USD and return you the local currency at an extremely inflated rate.
Market traders usually offer the best rates; sometimes these are called "black market rates".
Banks and money changers can offer better rates, if you are changing a significant amount of money; usually over $15,000 but this amount varies.
At the airport, rates are expensive vs. the local market.
ATMs will be more expensive than changing money at a bank or a money changer; because there are normally extra costs involved with foreign cards, the ATM network and even the bank to which the ATM belongs. It is best to check with your bank as they may offer reduced fees with some partner networks.
The worst rate you will get is if you walk into a shop and want to pay with USD (assuming this is common practice in the first place). Not only will you be losing out on the rate, you will be returned local currency as change.