(please contribute to the list with any other apps, and feel free to edit to make the list more accurate--this is a community wiki answer)
Notation for the list below:
With few exceptions, any app falls into on of these categories:
- it supports System Settings > Dynamic Type and Bold Text (I mark it as "Dynamic&Bold" below)
- OR also supports System Settings > Larger Text (I mark as "Larger Text")
- OR has its own app-specific font size settings ("App settings")
- OR has NO font size customization whatsoever ("No customization")
For each app, I also specify how its maximum text size compares to standard system apps at maximum Dynamic Type setting if different (the app font is Smaller or Larger than in system apps)--to make it easier for a given app to understand its text size limit, given only few minutes with built-in apps only at Apple outlet.
Social networks
Facebook:
- all UGC (News Feed, User Timeline, Individual post and comments): Dynamic&Bold (since 2015-Nov-06); much Smaller
- app interfaces: No customization; smaller
Twitter:
- timeline: App settings (3 sizes); smaller
- settings screen: Dynamic&Bold
Google+: No customization; Much smaller
Instagram: No customization; Much smaller
VKontakte (both iPad- and iPhone apps): No customization; Smaller
Email apps
Mail:
- plaintext emails: Dynamic&Bold
- HTML emails: font scales only with Larger Text, but not with Dynamic Type(!); but can be much smaller than for plain text emails / than in other apps
GMail: No customization; Smaller (both for plaintext and HTML emails)
Google Inbox: No customization; Smaller (both for plaintext and HTML emails)
AirMail: Dynamic Text, App Settings; no LargerText support (text gets smaller instead)
Calendar (by Apple): Dynamic&Bold
Task managers
OmniFocus:
- Dynamic&Bold throughout the app
- LargerText for leaf-level outline items and Note editing only; No for outline groups
Things: No customization; smaller
Instant Messengers
Messages (iMessage):
- history of chat with a contact: LargerText
- list of contacts on left sidebar: Dynamic&Bold
Skype: No customization; Much Smaller
Telegram:
- history of chat with a contact: Dynamic Type (no Bold Text); smaller
- list of chats: No customization; Smaller
Web browsers
Safari:
- regular browsing: No customizaion; pinch to zoom leads to horizontal scrolling. Smaller.
- has a special Reader Mode with App Settings: wide range of sizes, comparable to the range of Larger Text. Works only for article-type web pages, not everywhere.
All other browsers tried (Google Chrome, Opera, Yandex Browser, iCabMobile): No Customization; pinch to zoom leads to horizontal scrolling. Smaller.
Office documents
Google Drive: No Customization
Google Docs: No customization: only document zoom, leads to horizontal scrolling. Smaller
Google Sheets: No Customization: only document zoom, leads to horizontal scrolling. Smaller
Maps
Google Maps: No customization; Smaller
Apple Maps: Dynamic Type only
Other built-in apps
Settings: Dynamic&Bold
Notes: LargerText
Reminders: LargerText
home screen: No customization
built-in keyboard: Bold Text only
Other third-party apps
Pocket:
- list of articles: Dynamic&Bold
- aricle body: App settings: wide range of sizes, comparable to the range of Larger Text
Google Keep: No Customization. Much Smaller.