Most of my tropical backpacking experience comes from South East Asia. Mostly this was in Malaysia, Thailand, Laos, and Vietnam... all of which are considered tropical countries. I always advise people to travel as light as possible and to bring as few items as they can imagine in their pack. However, there are a few things that I think should not be relied upon to be able to pick up overseas.
If you have a favorite mozzie repellent
There are a lot of mosquitoes in South East Asia and even if you are not worried about the diseases they carry, you probably don't like to be itchy so it is vital that you carry repellent. These products are available in pharmacies all over the place but i discovered that many of them don't work and seem to be more of a fragrance than a repellent. I am a big believer in DEET and didn't notice many of the products even had that in any sort of satisfying percentage.
If you are fussy like i am about what kind / type of repellent you need, you should bring some with you because it might be difficult or impossible to find over there.
If you are like me, you have a particular brand or style of sport sandal (or maybe non-sport sandal) that you are attached to. For me this is Teva sport brand.
The problem you will face over in South East Asia are twofold: If you are a man you will find a very real difficulty in finding your size (I wear US size 13 and these are near impossible to find especially if you are looking for a particular type of shoe.) An even bigger problem is that a great deal of the merchandise that exists over in Asia are counterfeit. Sure, you might think you are getting a great price until your Tevas, which would normally last several years break after just a few weeks. It's not such a great value once that happens.
Suncream / Sunblock
It isn't that suncream / block isn't available over in Asia, the problem is that for whatever reason it costs about 3 times as much and you are often subjected to just "whatever the shop happens to have in stock." I don't know exactly why this is but I think a lot of it has to do with the fact that the locals tend to stay OUT of the sun as much as possible and you will have to search pretty far and wide to find a local that actually sits in the sun to tan on purpose.
I have seen standard sized bottles of good brand sunscreen going or nearly $30 US per bottle. Then their are brands you have never heard of and since this part of the world is so filled with copy items it is always a little bit of a concern to me that perhaps this product doesn't actually contain the SPF factor at all anyway. I have had this experience once and it was not nice as I sat in the sun with confidence only to get horribly horribly burned.
The lesser known brands are lessor known for a reason too. Many of the cheaper brands were not waterproof despite claims that they were and reapplication was so frequent that any savings on the purchase price was lost by it being necessary to use so much of it.
Undershirt travel pouch
It's an unfortunate aspect of backpacking: Theft is out there. It never happened to me, but it seemed like every new town that I went to there was at least one person on the hostel that had recently been pick-pocketed, had their purse grabbed by a passing motorbike, or had their bag rifled through while they were asleep on a bus.
It would take a truly bold thief or magician to be able to get something that is under your shirt and attached to your neck and closed with a zipper off of you without you knowing about it.
If you buy a quality one from a refutable brand such as Orbisey the thing is probably going to last forever. I know that I still have the one that i bought for $20 or so back in 1999.
Basically, I still think that you should leave most of what you think you need at home and travel as light as possible. However, there are certain items that you really should bring with you because it will be very difficult or impossible to find them once you are already over in the developing country that you are visiting.