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I now have 3 characters at level 75+, I am playing maps since a while (albeit only tier 4 or lower so far), but I have yet to see an Exalted to drop.
Is there some way I can increase the chances to see one? Or are they limited only to higher tier maps?
Exalted Orbs can drop randomly from mobs, chests, etc. from practically any zone; I don't believe they are limited to maps or other higher-tier zones.
The simplest way of increasing your chances to see an Exalted Orb drop is to kill more mobs, open more chests, and turn over more rocks.
You can also get Exalted orbs from certain Divination Cards, including Abandoned Wealth which drops in Arsenal and Vault maps, and The Hoarder which can drop in The Belly of the Beast level 1 and 2, The Harvest, and the Vault map. Farming those locations may ostensibly increase your chances.
I suppose it's also possible to get an Exalted Orb from the Divination Card Emperor's Luck, which gives 5x currency items, but it's not any more likely than it dropping naturally.
Exalted Orbs are also the primary currency used when trading high-value items. Learning what combination of attributes makes an item valuable, and then keeping an eye out for those items if they drop for you, may result in you finding an item you can trade for Exalts. Most players who have multiple Exalteds in their stash got them from trading for them.
I would recommend doing the following (this takes a bit of investment first though). It's what I've done in the past & also what worked out really well for me.
1.) You need a character who can tackle at least T10 maps without any big problems.
2.) You need a bit of currency to start with this, I'd say ~50c should be more then enough (only if you want to buy it all at once, and then run it over & over again, otherwise you don't need nearly as much).
What you want to do is buy "Mao Kun, Reef Map", and Sacrifice Fragments (Dusk/Dawn/Noon, skip Midnights, those are too expensive & don't offer anything the others do not, at least for our goal.)
The nice thing about Mao Kun is that it's a really, really, really good map to make good and fast profits, as most of your drops will either be raw currency or uniques (and uniques - for the most part - sell a lot faster then rares).
I don't know if you're familiar with the Map, but it's basically an island with a lot of pirates on it. Kill them as you pass through, and - once you find him - talk to "Fairgraves". After talking to him, all pirates die & respawn as ghosts. This respawn also applies to the previously killed pirates. Now you have to kill the ghosts. They have a small chance of dropping a "Booty Chest".
Once opened, the "Booty Chest" drops a lot of currency, and "Fairgraves" (as ghost) appears a few seconds after opening. He disappears after fighting you a while, and repeats appearing & disappearing until you've opened 7 "Booty Chests". After opening the 7th "Booty Chest" Fairgraves appears again, this time with ~15 ghosts at his side, and can finally be killed.
Now, regarding the sacrifice fragments (Dusk/Dawn/Noon). You have four slots in your map device. One of them is reserved for your map, and each sacrifice fragment added to the map device increases the item quantity (the stat that you want) by 5%, adding up to a total of 15%. Combined with the 30% of "Mao Kun", it means that you have a 45% increased quantity of items, which is fantastic for farming currency, and the "Booty Chests".
I ran this map very often in the "Breach"-league, and found it to be quite rewarding, and quite fast. It's much less of a gamble then "Putrid Cloister" is, and I've roughly made a reliable 25%+ profit per map run, not including rare currency drops like "Divine Orbs", "Exalted Orbs" or even "Kalandras Mirrors".
Small note: From my personal experience I would say that it absolutely isn't worth using chisels on "Mao Kun". Since it's unique you need 20 chisels to get it to 20% quality, and that isn't worth it, at least from what I can tell, since all you're effectively getting for that currency investment (a few chaos), is a 20% item quantity increase (45% -> 65%).
Regarding the estimated chance of exalted orbs while running this map: I can only speak for myself, but I spent the entire Saturday running this map (Ngamahu Cyclone Raider, 6L DD, no helmet enchant, Lv.(82)), and had 4 exalts drop in that time. This is really not what I would call a sufficient data sample, especially since it was my first time running it for a whole day, but getting an exalted orb on an average day should be the rule, not the exception.
From my personal experience (currently lvl 87 in Legacy clearing up to t13 maps), exalts drop rate increases as area lvl increases (this was to be expected and I am not sure that it is clearly stated by GGG).
With all chars I have leveled, I had 1 "guaranteed"drop until act 3-4 merc (sample of 5 different chars) and from there, RNGesus and probability theory is your friend. That means that you aim for high IIR/IIQ and Larger Pack Size, which is iften neglected by less experienced players. Larger Monster pack sizes increase drops which when combined with higher iir/iiq will yield more chances to drop exa.
